The pressure seems to be on mostly women (yes I know some men are amazing and do lots of childcare). It is women who are doing most of the school runs and if they need help they tend to ask other mums. Dads do not often get asked (yes I know some dads do all the school runs and pick up other people's children and are amazing).
When someone is having a baby the expectation is on the mum again to offer to help out and even give up their job to do childcare. Men on the whole are not on the receiving end of this expectation.
I went to a play group and saw many grandmothers who on talking to them were struggling with the expectation put on them. Some of them did not want to do childcare and others wanted to reduce it. I asked about their husbands if they had them and they said the expectation was not there for them and they didn't even get asked. (Yes some do but on the whole they do not).
Even MILs are criticised if they do not help but it is rare for a FIL who visits and does not contribute in anyway. (Yes some FILs are great).
Men don't hassle other men for childcare but women put pressure on other women.