Is the additional wage recorded by the monitor for the questioner's father, without the municipality's knowledge, considered permissible (halal), on the grounds that the questioner is helping his father in his work?

The permissibility of the work depends on its compliance with the municipality's regulations. If it violates the regulations, it is forbidden, and it is not permissible to take wages for these extra hours. The fact that the permit was issued by an employee does not mean it complies with the regulations, especially if the employee himself states that the municipality was not informed. The Permanent Committee has issued a fatwa prohibiting such acts, such as fictitious secondments or overtime, considering them treachery and illicit gain, citing Quranic verses that forbid this and prohibit cooperation in sin and aggression.