Sigh.
OP has clearly said she works less than half what her DH does.
Therefore, justifiably, the lion’s share of childcare falls to her. We would expect that of him if the roles were reversed.
But this means OP becomes the default parent because it is inevitable that the parent doing 80% of the childcare will be the one who remembers and carries out all of the finer details of that childcare - play dates, doctor’s appointments, birthday parties, being the person in charge of the ‘the schedule’, knowing their current shoe size, knowing what forms need signing for school and nursery…
Of course, you could do a handover and delegate parts of it, but like OP as a part time worker I find it is simply too complex and fluid as they move from stage to stage to hand over bits of it. For example, at DD school, 1 parent had access to their app. The app does EVERYTHING - meals, trips, clubs, reports, booking parents evening, homework.
If I am working part time and doing the vast majority of school runs then it HAS to be me who does this, because all that would happen if DH did it would be ‘remember, you need to take this to school today when you drop her off’ ‘remember to ask the teacher this at pick up’ ‘remember it’s mufti day tomorrow’. See how it wouldn’t make any sense?
But then inevitably school and everything in its orbit falls to me. I do the pick ups so I’m the one who has struck up school gate friendships and play dates.
As much as you all fall of yourselves to say your DH ‘does exactly half’ I wonder if that ‘half’ really just means he spends at much physical time with them as you do. I don’t really believe they organised the last birthday party, announced they were taking DC on an independently organised play date yesterday, or reminded you it’s dentist time and assured you he’s booked an appointment and has written it in the calendar.