Is a father's endeavor to build an additional floor for me, his son, considered a legitimate right that justifies paying money to a public official to circumvent building restrictions, or is this not considered a legitimate right?

If the prohibition against building another floor is for a legitimate and considered interest, then it is not permissible to violate the regulations, and the money paid to circumvent the law is considered a forbidden bribe. However, if the prohibition is due to obstinacy without a legitimate interest, and construction is only possible by paying money to some employees, then there is no harm in that. To rule on this matter, the government's objectives behind this prohibition must be known. It is advisable to present the matter to trustworthy scholars in your country. If they rule that it is forbidden, then inform your father gently, and if he is not convinced, seek the help of someone whose word he accepts, and continue to honor him. Enjoining good and forbidding evil with one's parent differs from others; it should be done through teaching without violence, abuse, or harshness in speech.