What should a family do after discovering their son stole the gold entrusted to them, and the father refuses to forgive them for falsely accusing him?

You have both made a grave mistake and committed a sin by accusing and insulting your father without evidence. You must hasten to repent to Allah, especially you (the female speaker), for your action is a great act of disobedience (uquq). You both must seek his رضا (رضا - contentment/satisfaction) and ask for his forgiveness by all possible means.