Is it permissible to take money for extra work without the partner's knowledge, in order to avoid family problems due to employing a relative who is not needed in the project?

It is not permissible to take anything from the company's money without the partner's knowledge. If you do that, you are betraying trust and consuming wealth unjustly. If the work does not require employees, then make that clear to your partner and do not accept employing his father without need. If you do accept his employment, then it is not permissible for you to betray trust and take what is not yours.