The most libertarian answer to the question of what it means to be a libertarian parent is that there is no answer.
Why do parents still read The Cat in the Hat to their children? The cat gives terrible advice, after all. His risk assessments are poor. He urges reluctant kids to break rules. His games are unstructured and seemingly pointless; "UP-UP-UP with a fish" is certainly not going to get anybody into college. He's a stranger who has broken into their house while they are unsupervised, bringing unsuitable companions with him. All in all, the book seems to cut against everything today's parents stand for.
Perhaps Dr. Seuss now functions as the Grimm Brothers once did, offering fantastical tales of transgressive horror swathed in comforting repetition. As in classic fairy tales—and even in their Disney adaptations—parents must be gotten out of the way before kids can come into their own. All the truly epic child heroes are orphans. For Seuss, it's enough to send mom out of the house on a rainy-day errand. Whatever magic the cat has, it is only possible in the spaces between supervision and routine, outside of the panopticon that modern parenting produces.
How can we make space for that kind of magic when cultural paranoia and increasingly invasive policies seem to require around-the-clock child surveillance and engagement?
The most libertarian answer to the question of what it means to be a libertarian parent is that there is no answer. A theory of the proper role of the state needn't dictate a theory of the family. There is no inherent hypocrisy in the idea of a libertarian who is a strict parent, for instance. There are as many ways to be a parent as there are to be a human being.
But libertarians tend to be more predisposed to certain ways of thinking about parenting. On the fundamental level, if you think the essential virtues are individualist rather than collectivist, you might seek to raise children who embody or share those virtues, as J.D. Tuccille does, by teaching them about the perils of trusting politicians and equipping them with practical skills that enable independence.