We have this at my DS's school. It's on a side road, so the residents got the road made a permit only zone.
The only exception is blue badges. I am one of half a dozen parents with one that is allowed to park on the road. But it doesn't stop other parents, taxis etc. parking along the road, blocking people in.
The blue badgers (lol) understand its a privilege and park at the end of the road, away from drives and drop kerbs, but the other parents just don't give a shit. With it being a new school year it's currently really bad. We have traffic wardens and PCSOs out periodically, but they sometimes contradict each other (the PCSOs told the blue badgers to park closer to the school, but that's where the driveways are) so we generally go with what the council appointed traffic wardens say.
If you have enough neighbours that are inconvenienced could you campaign to get the road made permit only? Does the school have enough land to expand the car park without encroaching on their field etc? Don't forget that both of these would be funded out of taxpayer money, so the council will want to go with the cheapest option.
Sadly it is part and parcel of living next to a school. I live right down the road, next to the secondary school and have an allocated parking space. But if I come home between 2:30 and 3:30pm there is always a twatty parent parked in there, usually in a BMW, Merc or Range Rover and I've been told to 'fucking get over it' when I've asked for my space back. I'm actually considering to have it rezoned as a disabled bay.