I disagree about not being 'on duty' whilst children are at school or being looked after.
Being a parent is an overarching responsibility that doesn't go away.
What working parent has not at some point done any of the following at work:
Worried about their DC
Paid for school dinners on the work computer
Telephoned or been called to discuss child related matters
Booked, cancelled, organised, etc. appointments for their children
Completed school paperwork
Planned out every minute from when you leave the building until kids are in bed...running through multiple school/childcare runs, squeezing in ballet, cubs, swimming and a play date
Stressed about having run out of fish fingers for said playdate.
Taken 'lunch' to go to a school event
Negotiated with manager or colleagues in order to get to parent evenings, sports day
Worried about how to make 20 odd days of 'holiday' cover Easter, Summer and Xmas breaks along with Half Term
Had to rescue via telephone the child that has missed the bus
And so on...
Some parents - often dads - do not do any of the above whilst at work. Men would often not get away with it. Women, are often grudgingly tolerated with their demand for greater flexibility and understanding.
And so, women continue (in general) to struggle to make all situations work.
I hate to say it, but I would say that in general a male working parent is likely to be a more of a part time parent than, for example, a singlefemale parent.
If I am correct in that assumption - barring obvious exceptions such as single fathers, or fathers that can and do combine work with the sorts of things on my list - then this would indicate to me that the crux of the issue here is still a feminist one. A man is more likely to be congratulated (often by women) for either 'tolerating' a SAH existence or for attempting to combine the roles of parent and employee.
It's bloody difficult for anyone, but for women it is even more thankless than for men whichever way you play it. Work or Home or Both.
It's about time parenting in general lost all the gender assumption and bias.