I'm okay with parents having power of attorney over their children (which, as I understand it, is the power to make contracts and things exactly like this in place of their children), and while a magic age is kind of a ridiculous band-aid, I'm not aware of any better method here (you can't exactly institute an adulthood test, I'd wager) for determining when a person assumes that power. Probably some power to issue curfews, curtail freedom of assembly (your parents can dictate who you can spend time with), and so on, because all of that falls under the general purview of being a parent. Other'n that, though, I'd really think children should have the same rights as adults, and in all of those cases it's the parents who assume control over those rights, not a government body.
Schools probably would need some specific exceptions (assuming some of those powers, but only for the time the child is literally at the school, and always subject to revocation by the parent) in order to function effectively, but even the current state of things is absurdly restrictive (see MSH's post). Naturally, this needs to be accompanied by an effective emancipation system and child abuse prevention system, but I don't know how effective the current versions of that are so I can't really comment.