Which of the parents has a halachic obligation to pick the religious institution a son or daughter goes to - the father or mother of the child?
The father is obligated to ensure his child's education (chayav b'chinuch), not the mother. He may do so personally or by hiring a school to supplement the home education (shlichus). Therefore, legally it is the father who would pick the school.
Of course, as a practical matter, both parents should try to pick a school they find mutually satisfying. For a husband to exercise his rights unilaterally may not contribute to marital harmony. In addition, mothers often have a better understanding of their child's needs and nature (bina yesaira) and have valuable insights to add to the process.
However, it is the father who is obligated in chinuch, not the mother.
An adult child may pick his own school and is not obligated legally to listen to his parents in choosing a school.