Yes they can.
I went through this before. In most schools there are limited parking spaces and if you apply for one; you must comply with "their" rules of usage.
They are not "forcing" on school parking, you are "requesting" to do so. Therefore they can ask/set any rules/restrictions they deem necessary for the "safety" of the students.
I feel it's blackmail to force written "consent" to search or you don't get a parking space.
Sucks I know, but from what I had found out it's legal for them to do.