What is the ruling on a father working in the tax authority, knowing that a fatwa from one of the shaykhs indicates the impermissibility of this work, and how does one repent from that and what are its implications?

Working in taxation depends on the ruling regarding taxes themselves: some are permissible, and some are forbidden. If the forbidden work is intended to alleviate and remove injustice as much as possible, then there is no blame, and its doer will even be rewarded. Sheikh al-Islam Ibn Taymiyyah mentioned that if the one in charge of authority strives for justice and removes injustice to the best of his ability, and his authority is more beneficial for Muslims than someone else's authority, then it is permissible for him to remain, and it may even be obligatory if no one else can do it. He is not held accountable for what he is unable to do regarding the removal of injustice, and he is rewarded for his good deeds. As for your father's earnings from forbidden work, if it was out of ignorance, then what has passed is his. If he knew and was negligent, then he should give the unlawful earnings in charity. As for the future, if he has a role in removing or alleviating injustice with the intention of striving for that, then he may remain, and this is confirmed if the alternative would oppress people. But if he has no role in removing injustice, then he is not permitted to remain.