Is it permissible to pay the zakat on my salary to my mother, who is divorced from my father and married to a man whose salary is low and insufficient for her and her children until the end of the month?

Zakat on salary becomes obligatory under certain conditions. It is not permissible to give zakat to one's mother because her maintenance is obligatory upon her child if she is in need, and her marriage to a poor man does not negate this obligation. It is permissible to give zakat to one's stepfather if he is poor.