Would informing one's father about the electricity bill, his using it to pay the amount, and then the mother's using the electricity to illuminate a room where people are backbitten be considered aiding in sin, taking into account the principle of "warding off harm takes precedence over acquiring benefits"?

Electricity itself is not forbidden, nor is it used exclusively for unlawful purposes. What the questioner mentioned falls under the category of harmful obsessive doubts (waswasa). The questioner should consider other means to help her family avoid backbiting (gheeba), and advise them with wisdom and good admonition. The principle of "warding off harms takes precedence over acquiring benefits" can be applied to ward off the harm of obsessive doubts. Exaggeration and delving excessively into religion are forbidden, as the Prophet, peace and blessings be upon him, said: "Beware of exaggeration in religion," and "The excessively meticulous are ruined."