My friend Jan calls them Wet Wipes, those superdads you see in the park, with the Scandinavian buggies that cost the price of a bespoke suit, the complicated slings, the strappy satchels filled with pre-dampened cloths, the muddy-puddle-ready change of clothes, the back-up beaker, the spare nappies and the sugar-free snacks.

Wet Wipes can be recognised by their thin, concerned smiles; their steely air of concentrated focus on the task at hand (fixing the handle on a microscooter); and their clumpy, ergonomic shoes. Look! There's one over there, on the other side of the playground, pushing a hi-vis-helmeted preschooler on a wooden, pedal-less bike. See that flat-white froth clinging to his top lip? Doesn't it make you yearn to borrow one of his pre-dampened cloths and politely but firmly wet wipe his mouth for him?

He is the male equivalent of the Tiger Mother, the ferocious, overambitious parent who has been introduced to the world over recent years in pop sociology texts from America (along with her sister, Sanctimommy). He makes sure to drop his presence on the school run into conversation; he lets it be known that he is hip to the new rules of conscientious dadding. He might sound benign, but his heavy tread resounds through contemporary male consciousness. Should I, too, be pushing for longer paternity leave and taking more of the strain at home? Should I be putting in 50%, instead of 30, or 20?

But back to my friend Jan: hard-working, harried, happy husband and father of three girls under six, and scourge of Hackney's Wet Wipes. Jan is a terrific dad – really a fun, playful, silly, loving, strong, tender, patient man. But he's not a Wet Wipe. He is not defined solely, or overwhelmingly, or even chiefly, by his child-rearing achievements and expertise. It's not his only mission in life to ensure that his daughters turn out to be the next Michelle Obama/Sheryl Sandberg/Jessica Ennis-Hill. He has a business, friends, an extended family, responsibilities outside the home, as well as interests separate from child-rearing, and he has a persona that is distinct from – but by no means in opposition to – Daddy.

I am not a Wet Wipe, either. My girlfriend, Danielle, and I have two kids. Penelope is four, Oscar just turned 18 months. I love them with an intensity of feeling I have not previously known, and the emotion increases all the time, as if someone were slowly turning a dial or tightening a vice. But one soldiers on. Just as one must soldier on, I suppose, even when the elder one requests a ninth consecutive reading of the Charlie And Lola anti-classic, I Am Not Sleepy And I Will Not Go To Bed on a Sunday afternoon, when the football's starting. One soldiers on at that point, I find, by surreptitiously transporting the sports section to the loo and slamming the door behind one in a huff.

Do I knowingly shirk my childcare responsibilities? Only the unimportant ones (enforcing chocolate ban, monitoring suitability of TV viewing, teeth brushing upstairs and downstairs), and only sometimes. Do my male friends do the same? Not naming names – there is still some honour among thieves – but certainly they do. Have we discussed it? Of course we have. Have we measured ourselves against those fathers more committed than us, and those less committed, and stood closer to the less committed ones to make ourselves look better than we are? Absolutely we have. Do we shy away from the Wet Wipes in the park and gravitate towards our fellow feckless fathers, who are doing their work emails and checking the football results instead of pushing their infant charges on the swings? We do.

When did being a good father get so complicated? Is there any middle ground, or must one either go full Wet Wipe or be a lazy, incompetent, dinosaur? Is it still possible, as it certainly used to be, to get away with the occasional omelette, some skewwhiff shelves in the spare room and, once in a blue moon, a full day with the kids so your other half can go out?

I know the answer to that last question. It's no, probably not. The expectations of fathers have changed. More is demanded of us.

My own father was and is a brilliant dad. He wasn't one of those remote, Victorian dads, who have to be introduced to their children when they turn 21. But he didn't change an awful lot of nappies, and if he'd ever offered to help with the housework, we'd have called an ambulance, or maybe the local TV news.

Are we any different, us younger men, his three sons? We are, I think. Both my brothers and I cook, clean, change nappies, often preside over bathtime and bedtime. So do our male friends. I do my share of the cooking; it's easy, I enjoy it. I help with the housework, although Danielle does more of it, and more of the childcare, too. And she does all of this while holding down a demanding job. In that way she is like many busy working mothers of her generation who find themselves stretched at work and stretched at home, simultaneously the beneficiaries of feminism's victories and the casualties of feminism's stalemates.

Feminism, no one needs me to tell them, is about equality at home as well as at work or out in the world. It's about duties, including childcare, being split 50:50. And that is fair and right and good. But it isn't what has happened. An equal split at home is easy to agree to, but difficult to achieve. A new report by the organisation of developed economies, the OECD, shows that on average British men spend 66 minutes on housework each day, while women spend 133 minutes cooking and cleaning. (Although the good news is that, according to the OECD, British women are the best in the developed world at balancing work, family responsibilities and social lives).

I can rationalise my behaviour if it would make anyone feel better: I could tell you that my girlfriend works four days a week and I tend to work six. Her job requires less after-hours entertaining than mine, meaning she is without fail the one who rushes home from work to look after the kids. And from the off, it felt as if nature and society had set us off on this path: she gave birth to them while I stood by, then she had almost a year at home each time, nurturing them while I carried on pretty much as before.

Of course, I could tell you all that, but let's be grown-ups: it's pretty much bullshit. I like work, I choose to do it, it gives me purpose, keeps me busy. (I am probably a workaholic, as is my father and as was his father before him, but that's another story.) I hugely enjoy the out-of-office fraternising that comes with my job, the foreign travel, the time spent out in the world. I moan about it sometimes, but only because it wouldn't do to constantly glory in my freedom.

If I am honest with myself, do I also detect a residue of unreconstructed masculinity, like crumbs in my pocket, that excuses all this by making me think that women, and certainly the woman I live with, are better at looking after small children, more practised, more suited to it? Honestly, I do. She is more patient and understanding and empathetic, but also firmer, more consistent – she spoils them less.

Also, she chooses her own path. She cherishes her time with the kids and yearns for more of it. When I asked her, while writing this, if she would rather I spent more time alone with the children, so that the childcare was split 50:50, meaning she would be liberated to work more, or socialise more, or sleep more, she said absolutely not: the thought of spending less time with the children rather than more would be horrible.

We are exactly equal, but we're not the same. And while our gender roles are not as rigid and defined as they were for our parents and certainly their parents, I know that my girlfriend still clearly performs more of what would once have been regarded as traditionally feminine duties than I do, while also putting out the bins. More than that, she feels the pull of home more than I do, and she is more conflicted about spending time away from it than I am. This is true for all our friends who have kids. It's not that the fathers don't struggle at all with this stuff, it's that the mothers struggle more.

In America, the current parenting buzz book is Jennifer Senior's All Joy And No Fun: The Paradox Of Modern Parenthood, which attempts to map the space between our expectations of parenting (domestic bliss, emotional fulfilment, photo opportunities, cute animal wallpaper) and the realities (anxiety, irritability, exhaustion, weight gain, higher vodka intake). An article on the website Salon recently located part of the blame for this supposed gap, for women at least, in the pernicious "good mother myth" – the impossible standards to which women hold themselves, the need to excel at work and at home, to be a brilliant boss, or employee, and a perfect mother.

And maybe that's the thing. If the Tiger Mother is the fearful figure most responsible for making working women feel they are somehow failing their children or their colleagues or themselves, then the Wet Wipe, I suppose, should be the guy most likely to make me and others like me – men who prefer to take a more freestyle approach to fatherhood – question our behaviour and doubt ourselves. And yet, rather than envy or resent or pity the Wet Wipe, I find myself ambivalent. He might regard fatherhood as a competitive sport, but his is not a game I'm interested in playing.

The children of Wet Wipes consider a banana an indulgence. (My own daughter considers a banana a tiresome but necessary prelude to an ice lolly.) Wet Wipe toddlers learn Mandarin and violin and probably mandolin, and for all I know already understand stuff I'll never manage to follow, such as string theory and credit default swaps and the plot of True Detective.

Each to his own, is how I feel. If some guy wants to take his kids on a character-building orienteering expedition (yawn) or a visit to the gardens of a stately home (double yawn) while I mostly slouch on the sofa with mine, making silly faces and scatological jokes, and giving them cheesy pasta and chocolate, and then buggering off back to work, so be it.

Someone – a woman – asked me recently if I worry about work-life balance? I realised that, while the mums I know would all say they are overwhelmingly preoccupied by that question, it had never even occurred to me to ask it of myself.

The key work-life balance question, inspiration for a thousand features in women's magazines, is: "Can you have it all?" I don't think most men, certainly not this one, ever thought we would have it all, or even wanted it all, if by "it all" we mean a seamless accommodation between a fulfilling life at work and home and out in the world. I am pleased that I have a hectic life at work and at home. I get stressed and frustrated and annoyed and knackered, of course, but I also revel in the chaos. It feels like being in the thick of something exciting and challenging and enriching. Of life, basically.

The aspirational images of excellence that women are presented with – she's a CEO, she's sexy and she bakes! – are far less attainable than the aspirational image of excellence men are presented with – he's a CEO! Which means we're not, on the whole, as stressed about failing to measure up.

Of course, this is all easy for me to say. Perhaps men like me don't want "it all", because we've long been able to have things as we want them. Having it all, for us, would mean no longer having just as little or as much as we want. Is it any wonder that even those of us who pay lip service to feminism still resist a full embrace of equality at home? An article in this month's Harvard Business Review, based on interviews conducted with nearly 4,000 American executives, male and female, suggests that however close to equality at work we have come, some men still regard family issues as primarily a female problem. When faced with work-life conflicts, the authors of the article report that men choose work without regret, because they see their main role as that of breadwinner. That mitigates any potential guilt about time spent away from home and children.

Of course, this is not true of all men. There are those who are willing to go all the way, to be 50:50 not just at work and in the world, but at home, too. But my sense is they're still in the minority. More men, I think, are like me: neither as brazenly unconcerned as the executives in the report, nor as "progressive" as the Wet Wipes.

It will take generations for true domestic equality to be achieved, so dramatic is the shift required. Oscar will do more childcare than I do, just as I do more than my father and he did far more than his. That is great, but it doesn't make me want to wear a papoose, or bake cupcakes, or invest in a new Scandinavian buggy instead of a good suit.

Andrew Solomon, author of the extraordinary Far From The Tree, about parents bringing up children with extreme challenges, recently made the following simple observation: "Raising children is terribly hard work, often thankless and mind-numbing, and yet the most rapturous experience available to adults." With the obvious exception of being hand-fed chunks of filet mignon by Scarlett Johansson's prettier sister in the bathtub of a private jet, I concur with the second half of that sentence.

While only a madman would argue that raising kids isn't hard work, I think "thankless and mind-numbing", for me, might be overstating the case a smidge. But then, as I say, I don't raise the children on my own. Far from it.

I see Penelope and Oscar every day, if often only briefly, except when I am out of the country for work. I would love to spend more time with them. The snatched, sometimes fraught conversations in the morning, when I'm trying to escape to work, and the testy jousting in the evening, when I'm trying to pack them off to bed, are not my idea of quality time, nor theirs. But I see them as much as I like at weekends, and yet I'm still relieved and appreciative when Danielle elects to take them to the park or on a shopping trip, or over to her dad's, allowing me to get on with my work in peace, or read the paper or slump in front of the Premier League rather than pigging Peppa Pig.

I leave prospective fathers – mothers, too! – with a small piece of childcare advice. One of the things about looking after little kids is that you might find that you have to wipe their mucky little mouths from time to time, because they are forever gorging themselves on sticky and melty things – or earth scooped out of plant pots. You can, if you want to waste money that would be better spent on fizzy beer for you and plastic trash for them, buy pre-dampened cloths to do this, which you can carry with you in a handy, resealable pack. Or, if you're the sort of fellow who can never quite seem to get it together – perhaps because, secretly, you don't want to be the kind of man who always has a damp hanky handy – what you can do is wipe their mouths on the back of your hand, then, when no one's looking, you can wipe the back of your hand on something else – grass, jeans, tea towel, child's shirt, doesn't really matter – and get on with your day. Works just as well as a Wet Wipe, costs less and, as long as no one sees it, it didn't really happen.

Honey, I'm home again: the rise of the look-at-me dad

David Cameron "Samantha had an away weekend recently… When you are on your own, it's a good reminder of how difficult it is to do anything else when looking after a little one, because you worry they might drown themselves in the sink while you're having a pee."

Nick Clegg "I actually like being with my children. I love having the opportunity as often as I can to take my children on the school run."

Ed Miliband "Being a good dad… weighs on me. One of the best things I do is take Daniel to school or Sam to nursery… my family are the most important part of my life."

Brad Pitt "One of the great privileges of being a father [is] you get to teach them things. Give them a little inspiration and the next thing you know, they're reading everything they can about a subject. Then they try and show you how much they've learned. It really is so beautiful."

Orlando Bloom "I haven't spent more than five days apart from him since he was born."

James Corden "How can you call yourself a dad if you don't [know how to change a nappy]?"

David Beckham "I'm a dad, I have four kids, I'm married, I do the school run every morning and make the dinner every night."

Simon Cowell "I never knew how much love and pride I would feel."

Prince William "I did the first nappy, it's a badge of honour."

• Alex Bilmes is editor of Esquire.