I want my two daughters, 6, to think critically, to fight for fairness and justice whenever they can. I want them to value equality above all else. But sometimes, I also need them to do what I say.This is pretty fundamental liberalism, the sort opposed not by "conservatism," but by the 17th century alternatives - divine right or patriarchal absolutism. This liberalism follows from the premises of our regime: our government is legitimate because we, as naturally equal individuals, consented to it. Because of our natural equality, it would not be legitimate for us to be ruled by other people, "to do what they say," unless we agreed to this arrangement, and we would never reasonably agree to an arrangement where we'd be ruled by the arbitrary will of another person, but only by the impartial will of a law that treats us all as equals.
Enter children. They do not consent to be ruled by their parents, or anyone. What right have we to rule them then, and to rule them by our arbitrary (parental) wills at that? Well, we can say, with Locke, that children "are not born in this full state of equality, though they are born to it," and must be educated into their birthright. Very well. And how do we educate children to their birthright equality? Here is where roads diverge. 21st Century Liberal Mother has taken an intuitively simple road: if the point of childrearing is to bring children into a state of equality and to teach them to see other individuals as equals, then the best way to raise them is by having them practice equality. Parents will treat their children as equals in the household (and teachers will do the same in school), so that children can rehearse the experience of adult equality without adult repercussions for occasionally slipping up and, say, dominating the playground occasionally. Because what is the alternative? If we raise children hierarchically, by treating them as inferiors to us whose own desires (to eat cake all day) matter less than what we desire and impose on them, then how will they ever grow into adults who understand and respect the principles of equality and consent that undergird our regime? And here I will offer the tempting dissertation spoiler for which you were all waiting: turns out that the experience of living under a hierarchical authority to which we do not consent as children is necessary to understand equality as an adult. We are better off without congruence between the liberal family and the liberal state, even though congruence seems more logically intuitive, especially to people who are not viscerally familiar with the fractious and willful nature of children.
So this woman seems to mean by liberal parenting what I would call "congruence parenting": - modeling your family life on the political principles of your state. But the distinction Phoebe is interested in - helicopter vs. free-range parenting - is a different one that I don't really see among this particular person's problems. This latter distinction turns more on our estimation of the goodness of children's uninstructed natures, whether childrearing should aim at curbing and shaping children's uninstructed natures or at preventing them from being curbed and shaped by outside influences. Helicoptering begins from the assumption that children left to their own devices will not become literate, disciplined, oriented towards the right ends, etc. So they need their parents to basically build them up from scratch into adults. Free-ranging assumes the opposite - that the methods of educating by adding things to a child like a school curriculum and music lessons and sports and basket-weaving is all going to corrupt and distort the authentic individual in the child, and turn him into someone worse. This opposition also fits uncomfortably into contemporary categories of liberal vs. conservative, since it arises from the late 18th Century opposition between empiricism/rationalism vs. romanticism. But there are both romantic and rationalist strains in both contemporary ideologies, so there's no way of easily identifying either helicoptering or free-ranging with the right or left. A cursory look at the history of American homeschooling in the late 20th Century will make it immediately obvious how free-range parenting can be harmonized with everything from fundamentalist Christianity to hyper-leftist secularism.
Finally, sitcoms that illustrate these tensions. The obvious one Phoebe overlooks is Gilmore Girls: a very young mother who raises her daughter as an equal, the daughter is growing up much more responsible and mature than her mother. One reading of this might be that congruence parenting works just fine, but only by accident, since it inspires such revulsion in children towards their parents' failure to act responsibly that they decide to shoulder all the responsibility themselves. But even there, we should recall that the arch-adult grandparents are brought back into this family at the beginning of the show, and they come at precisely a moment when the liberal parent is shown to fall short. Lorelai wants to send Rory to private school, but can't afford it. Responsible, sober New England WASP grandparents swoop in, pay for school, and insert their authoritative presence into the plot, creating a foil (and compensation) for Lorelai's lack of authority over Rory. Even though I always found Lorelai such an insipid character that I could never get into the show, it's a pretty good encapsulation of the competition between congruence and authoritarian parenting.