Generally, when the state attempts to restrict or terminate a parent's rights or responsibilities there must be due process of the law. Without due process it's an illegal taking.
The parent, not the state, is responsible for instructing his child. RSA 193:1, dating back to 1871, acknowledges that it's the "Duty of Parents." The state may require a parent to fulfill his duty to instruct his child; that is fine. The state may provide public school as an option for a parent to delegate this important task; that is also fine.
However, restricting this important parental duty without any due process of the law, i.e., without any evidence of a parent's failure to perform, and preemptively subjugating the parent to state regulation, that's an illegal taking of parental authority that needs to be addressed and corrected.