An international treaty -- which the U.S. Senate has not ratified -- could nevertheless be used to bar parents from homeshooling their children. An article at LifeSite explains:
The Home School Legal Defense Association's (HSLDA) Chairman and General Counsel, Michael Farris, warns that even though the U.S. has never ratified the United Nations Convention on the Rights of the Child, the convention may still be binding on citizens because of activist judges.
According to a new "interpretation" of what is known as "customary international law," some U.S. judges have ruled that, even though the U.S. Senate and President have never ratified the Convention, it is still binding on American parents. "In the 2002 case of Beharry v. Reno, one federal court said that even though the Convention was never ratified, it still has an 'impact on American law'," Farris explained. "The fact that virtually every other nation in the world has adopted it has made it part of customary international law, and it means that it should be considered part of American jurisprudence."
Under the Convention, severe limitations are placed on a parent's right to direct and train their children. As explained in a 1993 Home School Court Report by the HSLDA, under Article 13, parents could be subject to prosecution for any attempt to prevent their children from interacting with material they deemed unacceptable. Under Article 14, children are guaranteed "freedom of thought, conscience and religion" - in other words, children have a legal right to object to all religious training. And under Article 15, the child has a right to "freedom of association." "If this measure were to be taken seriously, parents could be prevented from forbidding their child to associate with people deemed to be objectionable companions," the HSLDA report explained.
The HSLDA report points out that
the U.N. Convention would:
 transfer parental rights and responsibilities to the [s]tate,
 undermine the family by vesting children with various fundamental rights which advance notions of the child's autonomy and freedom from parental guidance; and
 establish bureaucracies and institutions of a national and international nature designed to promote "the ideas proclaimed in the Charter of the United Nations'' and to investigate and prosecute parents who violate their children's rights. . . .
The State Will Determine the Child's "Best Interest'
Article 3: "In all actions concerning children," the courts, social service workers and bureaucrats are empowered to regulate families based on their subjective determination of "the best interest of the child." This article shifts the responsibility of parental judgment and decision making from the family to the State (and ultimately the United Nations).
There is hope, however, thanks to the U.S. Supreme Court's opinion in Pierce v. Society of Sisters (1925) (summary here, full opinion here). The summary:
Facts of the Case
The Compulsory Education Act of 1922 required parents or guardians to send children between the ages of eight and sixteen to public school in the district where the children resided. The Society of Sisters was an Oregon corporation which facilitated care for orphans, educated youths, and established and maintained academies or schools. This case was decided together with Society of Sisters v. Hill Military Academy.
Did the Act violate the liberty of parents to direct the education of their children?
Yes. The unanimous Court held that "the fundamental liberty upon which all governments in this Union repose excludes any general power of the State to standardize its children by forcing them to accept instruction from public teachers only."
In reaching its decision, the Court wrote that
we think it entirely plain that the Act of 1922 unreasonably interferes with the liberty of parents and guardians to direct the upbringing and education of children under their control. As often heretofore pointed out, rights guaranteed by the Constitution may not be abridged by legislation which has no reasonable relation to some purpose within the competency of the state. The fundamental theory of liberty upon which all governments in this Union repose excludes any general power of the state to standardize its children by forcing them to accept instruction from public teachers only. The child is not the mere creature of the state; those who nurture him and direct his destiny have the right, coupled with the high duty, to recognize and prepare him for additional obligations.
Pierce v. Society of Sisters dates from the era of Lochnerian (substantive) due process, which ended during the New Deal. But a new era of substantive due process may be upon us. Moreover, given the Court's present makeup (four conservatives, one waffler, and five Catholics) I am hopeful that a test of parents' rights would be decided in favor of parents. Given Justice Kennedy's quirkiness, I would be even more sanguine if Justice Stevens or Justice Ginsburg were to retire soon.
(Thanks to my daughter-in-law for the link to the LifeNet article.)