Actually OP I don't think YABU and I have been shocked by some of the reactions to your post. If parking around the school is unrestricted I can see no reason at all why the teachers can't use that to park their cars. If you say the school is currently undergoing building work then I am guessing that when many of the parents applied to the school it was because there was lots of space for the children to play in both in an organised, ie PE and games, and non organised way. I've lost count of the number of times I've heard teachers say that primary education is about much more than numeracy and literacy.
At my children's primary school there were computers in every classroom so I am at a loss to understand why some of the teachers on here have to lug laptops around. I work for a very large HE establishment, not as a teacher, but I do a lot of work at home. I take it home on a memory stick.
As for lugging two or three lots of 30 books, if there's that much marking banked up I would suggest that there may be difficulties with time management. If 90 pieces of work need marking in one night - can't some of the work be completed on paper and then stuck into the book to ease the burden.
The head at my dc's primary was obsessed with walking to school for children and families. 60% of her staff lived locally yet drove to work!
I also think the head's response was inappropriate. The head at my dc's primary was very similar, the school was in a very nice part of South West London. Many parents had choices and by year six year after year there were fewer than 50% of the original cohort left. Not because families moved away but because they had the choice to move to the independent sector. It is not the way for a school to be part of the local community and regrettably the one I'm thinking of only made such comments because of the ginormous chip on her shoulder.
Final point, once one of the teachers' cars parked in the playground gets scratched or dented, how much longer will the children actually be allowed to play freely at playtime. It's a playground not a car park IMO.
This school wouldn't possibly be expanding from 45 to 60 would it?