If you have joint legal custody of your child then you must get the other parents permission and written consent to change the child's school.
The school will usually not allow the child to change schools without the other parents permission if the parents are separated and have joint custody of the child.
If you have single custody of the child then you don't need the other parents permission to change the child's school.
But if you're not separated and are still married and there's no court ordered custody of the child then you can change the child's school without the other parents permission.