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Yes, unless the child is emancipated. Until a child reaches 18, the parent is legally responsible for the child. Therefore, the parents have complete authority to order the child to do anything that is not illegal.
However, if the child becomes emancipated (legal process in which the child requests the court to remove his/her minority and be considered an adult for legal purposes), he can do as he/she pleases.
One caveat is that the child still has to attend school - at least through high school.
If it goes to court and the judge rules that a guardian should be awarded the children, the parents have no say in who that person is.
There is a limited emancipation, that fully emancipates the child, except for financial support, which can be ordered to continue from the parents.