Irrelevant. They are still the parents, the school is not.
But yeah schools, any school have the right to establish codes of conduct which includes how a student may dress while on campus, provided such code does not violate a students right.
Not public schools.
You are wrong and there have been court decisions that explain why you are wrong. Perhaps you should become familiar with the term “In Loco Parentis”. Which though not the force it once was in education has not been scuttled entirely. It still applies to dress codes.