We also had this problem at DS school. The repeated gentle reminders were ignored and eventually the residents had the road made permit only.
The local police and parking enforcement agents allow blue badge holders to park there still as long as they don't park over driveways. Taxis are allowed too as they're dropping off.
The problem is, the blue badge thing hasn't been well publicised. I regularly park there with my badge, at the end of the road with fences, no houses, so I'm not blocking anyone. Other parents see this and decide they'll park along that road too. Going back to my car one morning I saw the traffic warden go past my car and start writing a ticket for the person parked in front of me.
The woman was actually walking behind me and started yelling at the warden to stop, then noticed I didn't have a ticket and started yelling why not. The warden said why and she started going on about how unfair it all was. I've had looks and moans from other parents too.
Another mum was recently moaning on Facebook that she'd been 'abused' by a teacher for parking in the car park even though this has always been forbidden. She has a massive range rover too.
So even with a CPZ parents park where they want anyway, there isn't alway anyone on duty and the taxis park wherever they like.
We also live next to the secondary school and our flats have their own little car park. Parents are always pulling in to drop off, not using a space, but just pulling in to be off the road so they can drop off their darlings. They often block us in and the drop off is never as quick as they think - the kids don't jump out immediately, it appears to be having a chat/arranging when/where to pick up, kids get stuff together, goodbyes, kids get out, kids go to boot and start sorting out bags/pe kit/cookery stuff, then door open again to ask a question/say bye again, then parent watches them leave and waves, then parent does a 65 point turn to get out of car park and then I can finally move. Getting out of the car, evils, beeping, asking politely if they'll move gets either ignored or told "it'll only be a second" or at its worst "fuck off, love".
Boils my piss, but I'm aware that to others I look like part of the problem.