I suggest that moral responsibility for training and education of children lies firstly with the child's parents, although not as part of a package of responsibility attaching merely to having caused the child's existence. That Objectivist position focussing on causal relationship, down from the era of N. Branden in the 1960's, was off the mark. Moral responsibility for training and educating the child lies firstly with the child's parents, I suggest, because of the moral goodness of responsiveness to persons and the potential person they may become, responsiveness to persons as persons.
That responsiveness is, I say, the core of moral relations among people (and indeed, differently, relations of a self to itself). That is the preciousness that is the moral in a social setting. This position is a cashing out of the concept of moral justice, treating a thing as the kind of thing it is—that moral virtue. What a thing is includes its internal systems, but as well its distinctive external relations, actual and potential. The relations of responsiveness to persons as persons have a specially intense and distinctive character in the relation between the persons who are parent and child (natural parent most strongly, of course, but strong with adoptive parents as well).
Additionally, there is a moral goodness in the benevolent protectiveness—that responsiveness—between any adult and any child. That such responsiveness fosters continuance of the species human as human may well be the underlying biological reason for this responsiveness. But that is not the reason the responsiveness of parent or other adult to the child and responsiveness of the child to them as persons is moral. Rather, the nature of value in the life of individual humans together, which is their best situation in the world, is the source of the moral goodness of such responsiveness to persons as persons.