Is it permissible to pay a monthly salary to the mother and sister from the company the questioner works for, given their difficult living conditions, low income, and numerous debts?

What matters is the fulfillment of the conditions for entitlement to the salary stipulated by the granting authority, without deception or trickery. If the real reasons are stated and the authority agrees to grant the salary, then there is no objection. However, trickery and deception are not permissible, due to the Prophet's (peace be upon him) saying: "He who cheats us is not of us," and his saying: "Muslims are bound by their conditions."