Which case determined that parents could educate their children in a manner consistent with their beliefs?

The case that determined parents could educate their children in a manner consistent with their beliefs is Wisconsin v. Yoder. This landmark Supreme Court decision, issued in 1972, recognized the rights of parents to direct the upbringing and education of their children, specifically in the context of the Amish community's choice to withdraw their children from public schooling after the eighth grade in order to preserve their religious beliefs and lifestyle.

The Court held that the state's interest in educating children must be balanced against the parents’ rights to exercise their religious freedom. It ruled that the First Amendment guarantees the free exercise of religion, which includes the right of parents to educate their children according to their faith. This case is often cited in discussions about parental rights and religious education in the United States.

In contrast, the other cases mentioned do not address the specific issue of parental rights in education. For example, New York Times v. United States dealt with freedom of the press, Roe v. Wade focused on a woman's right to choose regarding abortion, and Tinker v. Des Moines addressed students' rights to free speech in schools. Each of these cases contributes to the broader understanding of constitutional rights but does not pertain to parental control over educational methods as Wisconsin v. Yoder does.