I think that you are not adequately characterizing MNR’s position here. In the link you referenced, Rothbard writes:

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Mom and Dad: Hillary is Out to Grab Your Kids! Hillary is the prophet of the children’s “rights” movement, a movement now openly backed by left-“libertarian” philosopher Tibor Machan, a movement that encourages 11-year-olds to sue their parents for “malpractice.” Any parent can be accused by some officious biddy of “malparenting,” and since 11-year-olds and 9-year-olds and 5-year-olds are not exactly legal beagles, you know darned well who will really be doing the suing: leftist ACLU-type lawyers, lawyers cut in the mold of Hillary and Thomases. When the campaign began, ultra-left social theorist Garry Wills hailed the “brilliance” of Hillary as a “children’s rights theorist.” That means: the government, the leftist lawyers and social workers are out to get your kids! There is a lot of confused discussion about family “values,” about what these terms really mean, and about what they don’t mean. Well, there’s one clear test: “family values” means that kids get brought up, get governed by, their parents. Anti-family values means that other folk; bureaucrats, lawyers, duly licensed social workers and counselors and “therapists,” the rapacious, power-hungry, leftist New Class, get to bring up and run everyone’s kids: all in the name, of course, of children’s “rights” and “liberation.”
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I do not take this as Rothbard repudiating his earlier position, or supporting parental coercion. I think that he is, however, advocating that lawyers not take advantage of the right of children to run away, as a means to bolster government institutions and undermine family values.