One way forward might be to identify every site - churches, village halls, shops, pubs, playgrounds, small industrial estates etc etc - with a car park within the school area, and to really, really heavily push 'park and stride' using partnerships with these sites.
Minimise, or prevent, car parking next to the school except for disabled pupils / parents and for staff (who will generally be there much earlier, and stay much later, than drop off / pick up) but ensure that there are enough spaces in the sites a short distance away. I have seen school areas where the rules for on-street parking in bays etc exclude 8-9.30 am and 2.30 - 4 pm, although it is fine outside those hours - so residents and visitors can use on-street parking overnight or during school hours, but there is no parking at drop off or pick up times.
IME, school gate car parking is more staggered, and less of an issue, at schools which have onsite before and after school childcare at a reasonable cost, as more working parents, rather than dropping children off at 8.25 [if the gates open at 8.30] and haring straight off to work (and then reversing this at the end of the day), stagger arrival / departure times by using breakfast / after school clubs.
Possibly worth thinking through how families with different needs - those needing to drive to work from drop-off, those with mixed age siblings, those who are disabled or have a disabled child, those who could walk but don't - might need / want to use parking, and think not only of rules / physical provision, but also incentives to behave around parking in a way that works for the school and its neighbours.