Troubadour has a more useful point, namely, that the Rothbardian concept of rights of children may cause legal "child prostitution."
The mother, then, becomes at the birth of her child its "trustee-owner," legally obliged only not to aggress against the child's person, since the child possesses the potential for self-ownership. Apart from that, so long as the child lives at home, it must necessarily come under the jurisdiction of its parents, since it is living on property owned by those parents. Certainly the parents have the right to set down rules for the use of their home and property for all persons (whether children or not) living in that home. (Ethics of Liberty, 103)
In writing this, Rothbard, too, confuses (1) in the previous post with (2). The fact that parents can set rules for using the house does not give them dictatorial powers over any child who lives there. Moreover, a family is not a political community.
Children are different in the sense that if Smith throws out Jones his adult guest, then Jones will go elsewhere. The child will die. So, if Smith says to Jones: "Either you let a bunch of homosexuals rape you, or you leave my house," then Jones will be unperturbed. He will simply leave the house, thinking Smith rather rude. But a child cannot do that. Therefore Smith the parent can threaten the child with expulsion followed by death, unless the child submits to being raped to Smith's profit.
If the parents have the right to abandon a child at any time, killing him in the process, then a fortiori, they can offer him a choice of the lesser of the two evils of abandonment vs. prostitution.
Suppose the parents did decide to abandon the child and let him know that. The child, facing the grim prospect of death out in the hostile world, makes his parents an offer: "I'm more valuable to you alive than dead. Instead of throwing me out, let me make money for you by being a child prostitute. I'll work for food and shelter. Invite some men over and let them bang me, and I'll serve you well." The parents mull this over and agree. Is this a reasonable situation?
The existence of formal social abandonment procedures a la Walter Block (which in themselves are a wonderful refinement of Rothbard) is beside the point. In practice, this problem is very unlikely to be widespread. But the theoretical thorn in the libertarian's flesh does exist.
Or in other words, abandonment procedures are not enough. There have to be rules for what sort of things the parents are allowed do to a child that in badness come close to abandoning him.