I think women get the short end of the stick in this scenario.
They get the pregnancy pains, the delivery or surgery, birth injuries, the endless breastfeeding and every thing that can go wrong with that, the multiple wake ups, the 4 month sleep regression, the loneliness of having only a potato with eyes for company day in day out....
A few months later when baby is sleeping most of the night, in a more predictable routine and doing interesting and cute things during the day, Dad swans in and gets to take over while a chorus of people tell him how wonderful he is for doing it.
Meanwhile a knackered woman who might still be waking up at night to breastfeed, returns to work and gets to spend her breaks sitting alone in a storage closet with pumps squeezing her breasts.
I know all babies are different, but my newborn experience was a hellscape. From 6 months onwards my baby has been an absolute delight. It would have really pissed me off if I'd done all the hard work early on only to have DH come in and do the easy fun part in the name of 'equality.'