What should the father do regarding the years in which he erroneously paid Zakat, and the years in which he did not pay Zakat at all?

Firstly: Zakat (definition) is not permissible before possessing the nisab (definition). What is given before that is considered a voluntary charity.

Secondly: Whoever neglected to pay Zakat in the past is sinful and must repent and pay Zakat for all the missed years. He must accurately calculate it to the best of his ability.

Thirdly: It is permissible to pay Zakat in advance after possessing the nisab, before its due time.

Fourthly: Zakat is due according to the lunar calendar. It is not permissible to rely on the Gregorian months. Whoever did so must pay the Zakat difference along with repentance.

Fifthly: If the father paid Zakat before reaching the nisab, it is a voluntary charity. If he paid it in advance after reaching the nisab, his Zakat is valid. If he delayed it according to the Gregorian year, he has erred and must repent and correct the mistake. The default is that your father's liability is clear, and he is not obliged to do anything except with clear evidence. If he is truthful, his word is accepted.