When a profile update form is assigned to a student, there will only be one copy of the form. If their parents live at two different addresses, then two different Household blocks will appear on the form, and either parent can fill out the information.
You can require zero signatures, one signature, or signatures from all parents. If you require only one signature, then then first parent to sign the form will submit the form. The second parent will still be able to view and print the completed form.