Hello. Thanks for contacting us.
You raise an interesting question. In theory, a parent can do so, but the other parent can give permission. In such a situation, the school need only have permission from one parent.
This assumes the parents are not divorced, and the parent with primary custody is not in favor of the extra-curriculars.
Parental rights to control children tends to fade out as the child approaches adulthood.
But if you are talking about actual children, who can never be emancipated before age 18, then parents have rights to control their extra-curricular activities (assuming, again that there is no reason the parental rights don't exist -- such as a court-ordered guardianship).
This is the general information, anything would be particular to a situation -- and always subject to court order, if the court believes the order is "in the best interests of the child." Its rare, but courts can intervene for that reason.
Please let me know if you need any clarification.