Parents do not own their children, but are agents for them. Unlike persons of majority, minors (which includes all incompetent parties, such as children, the mentally ill, and anyone else unable to care for themselves), hold extra rights. They have the right to be taken care of by others.
It's not a matter of the libertarian parent saying, the moment their kid is born, "Okay, I've done my part. Now, get your ass out there and start earning a living!" No--by bringing a human being into this world the parent has, by their action, entered into an implied contract to care for that child. This care includes not allowing the kid into situations where harm might come to it.
For example, the parent could not say to some sadist, "Sure--for a hundred bucks, I'll let you spank my kid." Even if the kid agrees to this, it would still be wrong, as it might bring serious emotional or physical harm to the child.
The age of majority varies from child to child. For some, it might be 15; for others 18. Some are ready at 10, while others might not be ready at 20. But, the law must pick some general age at which most should be considered ready to handle themselves in society, and 18 is the general age picked. (I don't care for the fact that in the US this age of majority is sort of dribbled out: a little here, a bit there. At 17, you can drive, at 18, you can smoke and vote, but you can't drink until 21. Just pick a freakin' age and stick with it!)
But, in every state a child may apply to the court for early majority. Howard Hughes gained his majority at 19, and others have done so much younger. In other cases, some people never gain their majority; for instance, the severely mentally retarded. You can't reasonably say to someone born with an IQ of 50, "Well, you're 21 today. You're on your own!"