Historically, children were considered the property of parents. Under the scheme of patriarchy wives and children are under the control of the family head, the husband of the family. Much of this hoary antiquated thinking is still promoted by far right conservatives and the vestiges of ancient thinking die hard.
Today, children are persons in their own right and modern progressives recognize the personhood of children. Children have the right to grow up free of debilitating dogmatic superstitious thinking. Instead parents and communities must encourage children to think critically and determine their own path through life to suit themselves.
Authoritarian parents view the autonomy of children as threats to their antiquated way of life. Anachronistic groups like the dominionist inspired and led Parentalrights.org want to seal off the family from the state to avoid heretical ideals like children’s rights or the emancipation of women creeping in. They advocate a constitutional amendment to protect what they call parental rights, but are really restraints aimed at the protection of the antiquated system of patriarchy. This is why they emphasize the taboo of non-interference in family matters (a strong meme). The goal is really to keep patriarchy going and ward off any progressive moves that would liberate wives and children.