Indeed parents have a duty to their children. Clearly this cannot and should not mean that they're allowed to treat their children however they want or that they're allowed to educate them on what and how they want. There can be a space for them to make decisions but the rights of the children always have to take precedence.

It's only natural for the government to step in and determine exactly what this means. Allowing the government to do this without limits is problematic, so you balance that out with rights assigned to parents.

It's not an easy question to determine where the line should be drawn but I don't think it's really all that totalitarian to prohibit home schooling. It ensures that children are educated, that they socialize with other children, get confronted with different points of view and gives them the freedom and ability to safely explore the world and themselves in an environment where there parents have much less influence over their lives. It should not be forgotten that not all parents are included to provide that without force.

If you could prove that homeschooling is unhealthy or even worse than public schools, then you could have a point. But now you are just stating your opinion and suggesting that your opinion should be reinforced by the government.

The data shows that homeschoolers perform better than public school students not only academically, but also on measures of social, emotional and psychological development.

Parents may or may not make the best choices for their children; there are no guarantees there. When talking about government, though, it is guaranteed to make poor choices, and self-serving ones to boot.

What defense do free citizens have against the ambitions of a government, when they don't have the right to pull their children out of its clutches?

Then every society is totalitarian. The fact is that society already tells you how to raise your children and will remove them from you if need be.

There's a wide, wide range between (e.g.) beating your children and refusing to hand them over for state indoctrination. Pretending otherwise isn't a reasonable argument.