Is illegal work and unauthorized stay forgiven, if a US citizen applies for their parent in the US?

Generally, if a foreign national parent entered the U.S. lawfully and with inspection, his U.S. citizen adult child successfully may petition for him to adjust status to become a Lawful Permanent Resident (to get a "Green Card") regardless of whether he may have worked without authorization and regardless of whether he may have overstayed his initial authorized stay. If the parent entered the U.S. unlawfully and without inspection, however, then a more complex analysis is needed in order to determine eligibility. Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.

Your father's unauthorized employment will be excused.