who has more power?

I have gotten into a few discussions with my roommate about this idea. The conversation centered on the use of cell phones in schools. Her argument is that they schools don't want students to cheat on tests by texting each other answers to said tests. which i suppose is valid. however i also have, in my opinion, a much more valid reasoning. The parents get their kid a cell phone and tells them to keep it on their person at all times...but the school says you cant have cell phones on you so you cant bring it or if you do you have to leave it in your locker. now as a parent (which I'm not but still and wont ever be...hypothetical situation) i explicitly told my kid that they are to keep it on their person at all times. Who trumps who? in my opinion as the responsible parent my word is to be obeyed above the schools. they don't have legal custody of my child...i do. their job is not to police my standards or what i tell my child to do.
same thing if i had a car. if you had a license and a car you could only drive it to school if you were a senor. that was the school rule. wait a minute here...something seems wrong. The license is mine, i own the car, i paid for the car, i pay for the insurance and have STATE permission to drive it, I'm say 18 so i don't need dear old mom and dads permission...but somehow the school can tell me i cant drive something they don't own and pay for? how does that work? some one please tell me what the school system is thinking!!!