Two points:
1) “Compulsory education” puts compulsion before education. Before anything else, schools which assemble their clientelle through compulsory attendance statutes teach children that strangers will command their time and that their parents will be powerless to prevent this. I can imagine a few lessons more destructive of families and of the values of a market-oriented democracy, but not many.

2) Often, the words in which people express their arguments predispose them to their conclusions. For example, “mass transit” means literally how lots of people get around. The term could apply to bicycles and roads or to shoes and sidewalks, but proponents of centralized transportation systems pull a bait-and-switch to leap from an agreement that a “mass” of people must “transit” from home to work to the conclusion that some city or other needs a centralized transit system. Similarly, defenders of the NEA/AFT/AFSCME cartel’s exclusive position in receipt of the taxpayers’ K-12 education subsidy pull a bait-and-switch when they confuse “education” with “school” and “public education” with “government-operated schools”. Part of this confusion relates to a deep ambiguity of language. “Somewhere in Hawaii a woman gives birth every five minutes—and we have to find that wahine (woman) and make her stop!” is a comment on syntax which probably has parallels everywhere. Even with their precise notation, mathematicians have to pay attention. “For all X there is a Y such that F(x,y)…” “For all” really means “for each”.

“Parents should control the education of their own children” still leaves open the possibility that the mechanism through which parents exercise this control requires them to do so through collective action. Who composes the resolutions on which parents vote? When are the Board meetings? Who holds the gavel? Who counts the ballots?

Why suppose that collective decision-making processes will yield results which better represent the preferences of individuals, collectively, than will market mechanisms? What does it mean to aggregate the preferences of individuals? What would Americans eat if we voted on each day’s breakfast, lunch, and dinner menu? How many people would prefer the result of such a process to the result of the current market-oriented mechanism?

My response to advocates for universal health care: “I didn’t know the universe was sick.”