It's quite funny all the posters laughing at jsmoke when it's actually a good question.
I'm not sure it's a good question.
Parents (also guardians) are responsible for the children in their care, and liable (up to a point) for their actions.
The role is plainly that of guardian/custodian/provider/tutor/coach (etc). Which are obviously not legal terms as IANAL.
But then the state can and does tell parents what they may and may not do with/to their children. Eg "your children must go to school" and "you must not beat your children". Etc.
But that does not indicate property or ownership. It's just a common framework for society. The state also dictates what you may/may not do to other adults - whether of your household or not.
Clearly the state doesn't own anyone. I'd say your employer in most cases "owns" us more than the state does. And even that is just a figure of speech.
If anyone thought the state owned their children or indeed owned them they'd be trying to escape, and in that case we'd be the kind of crackpot country that built a wall to stop them (escaping)

Imagine NK. We're not NK.
Our
society does control us for sure. But it's not directed by the state or some committee

We have collectively established societal norms, and we consciously or subconsciously value and encourage conformity. We try to be "normal" and we gravitate towards "normal" people - who play by the rules. We're *supposed* to have kids (or you're not
normal). We're supposed to want loads of stuff and to have credit cards and loads of personal debt. We're supposed to have a nice family car, nice clothes, etc.
None of this is being directed by the state - but by societal conditioning which we all contribute towards creating.
Alternatively, all of the above is crap and I'm bored.