Per Ilya Somin, there might be: “In two landmark cases in the 1920s, Meyer v. Nebraska and Pierce v. Society of Sisters, the Supreme Court ruled that the Due Process Clause of the Fourteenth Amendment protects parents’ and guardians ‘to direct the upbringing and education of children under their control.'” Meyer struck down a ban on instruction of students in foreign languages before eighth grade, while Pierce struck down a ban on private and religious school education. While authorities presumably have wider leeway to regulate pedestrian activity on public streets than instruction that may take place within private homes, churches, or schools, a degree of regulation that forcibly substitutes the state’s judgment for parents’ on debatable issues of child-rearing might cross a line.
The idea of a Constitutional right of parental autonomy appears to be alive and well on both conservative and liberal sides of the Court, but some may be surprised at which current Justice has written most critically of the idea: Antonin Scalia, because of his dislike for “substantive due process” theory and in general its protection of individual rights not enumerated in the Constitution. In a 2011 article I haven’t had a chance to read, David Wagner traced the Scalia-Thomas conflict and apparently also looked at whether Scalia continues to count as a holdout given what might be a softening of his views on the issue.