In practice parents and the government agree most of the time because it's established science that no one contests, and cases where the government got it wrong are a large class of exceptions. Then people see the government forcing its rules even when they're in the wrong and lose faith in the whole system, including the general consensus. Then you get more counter-culture types fooling around with weird plants.
Whereas in the absence of coercion, the people who are right when the government is wrong are able to openly prove it without being punished, which allows the government to improve its recommendations more rapidly, which reduces how often their recommendations are wrong and thereby the number of people disregarding them wholesale after watching them be incompetent pigheaded bullies.
Or corporal punishment, where scientific consensus since 1990s is that it is harmful. The countries which have banned it since did so ahead of change in public opinion.
There are times when the government is wrong, but it is by far more right than wrong. Thanks to government intervention we saw childhood mortality drop from the high levels in previous centuries, and letting parents just do their thing would eliminate much of that progress.
Keep in mind: This whole "debate" is about parents doing insane crap, like homeopathy and "natural remedies" for which there is no positive evidence, not debatable stuff where there's honest evidence for multiple positions. Parents failing to provide proper medical treatment for their kids is abuse, just as it would be if your boss somehow forced you to drink bleach. And, yes, some of the "natural remedies" are bleach:
https://en.wikipedia.org/wiki/Miracle_Mineral_Supplement
> Miracle Mineral Supplement, often referred to as Miracle Mineral Solution, Master Mineral Solution, MMS or the CD protocol,[1] is a branded name for an aqueous solution of chlorine dioxide, an industrial bleaching agent, that has been falsely promoted as a cure for illnesses including HIV, cancer and the common cold.