> It is also one of the last tools that prevents parallel societies from forming.

Consider German history. They had the Nazis in power. Half of Germany had the Communists in power. So, you don't want Nazis and Communists home schooling their kids - you want those kids in regular school, where they can be taught about democracy and human rights. But if the Nazis or Communists are in power, you absolutely want to be able to homeschool, rather than have that garbage beat into your kids day after day at school.

So given their history, I can absolutely see why they should be afraid of home schooling. But I can also see why they should want the freedom to do it.