I think you are right to claim that parents have responsibilities towards their children, and do not have the right to raise them "any way they want." Children are not property. The larger moral concern, however, is that the state will decide what children are to learn, and in American society, we are most fearful of that (because of our history with totalitarianism and communism). The law protects the rights of individuals. So parents have a legal right to withdraw their children from state-sponsered education. They also have a legal right to teach them rubbish. However, I would argue that they do not have a moral right to teach them rubbish, particularly if it is rubbish that is harmful (Santa Claus probably does not fall into that category; but Abstinence is harmful rubbish).
Read another response by Miriam Solomon
Why do parents have the right to decide anything about a child's upbringing, or their moral, social, political and spiritual education? Young children are trusting when it comes to their parents, and may even believe falsehoods if their parents are the ones who are repeating these falsehoods. So why do we recognize a unilateral right for parents to teach their children whatever they want, and to withhold whatever information from their children that they deem appropriate? Why do we let parents pull their children out of sex ed class, or teach them a religion as a unilateral source of truth? Shouldn't parents have responsabilities, instead of rights? Surely shaping a child's mind, personality and outlook is not the "reward" parents get for feeding and clothing them! Is this just a practical issue ("There's nobody in a better position to take care of the kids, and there's no way we can stop people from teaching them whatever they want")? Or is there some fundamental moral reason parents have the right to do what they want with (or to) their children's minds?