If anyone knows about the legal right to homeschool, that would be very useful information to post on CathInfo.
Through the sacrament of Matrimony parents have the responsibilty to educate their own children, and the grace to do it.
They may delegate that authority if they are unable to fulfill the requirements. They have the responsibility to delegate the authority to somebody who will educate them in the faith. Education is a waste of time if it is based on false creed/s, such as happens in almost all Church and state schools.
Parents also have the responsibilty to protect their children's purity and innocence, and this is impossible where so-called "life education" which goes under various names, abuses their innocence and interferes with their growth, and may drive them into a life of sin.
There are Vatican docuмents which spell this out which I cannot name right now. Maybe somebody else has them at their fingertips.
As far as state law goes, I believe that the law changes from state to state.
But God's law has supremacy. The rights of the parents stem from natural law, which is based on the natural order, as God created us.