In principle, no difference exists between the degree of personal sovereignty possessed by a child and an adult. Practically speaking, however, dependence on adults limits a child's sovereignty. Dependence implies a degree of immaturity and a limited capacity to employ reason in making choices. Thus, a parent or parent surrogate has a right to be involved in the educational choices of a child. However, there remains a question about the nature of that involvement.
Parents can attempt to influence educational choices by a child in several ways.
First, a parent can use force to impose educational choices on a child. Second, a parent can use threat or intimidation to impose educational choices on a child. Third, a parent can use contrived incentives to influence a child's educational choices. Finally, a parent can use persuasion to influence a child's educational choices. Clearly, the first two options are coercive and inconsistent with a Libertarian philosophy.