Is it permissible to supplicate for a mother who abandoned prayer, specifically with the noble verse: "And say, 'My Lord, have mercy upon them as they brought me up when I was small'," and is this supplication valid only during her lifetime, and is it permissible for the father as well?

It is permissible to supplicate for a living disbeliever with mercy, but the more correct and better approach is to supplicate for their guidance. As for a deceased disbeliever, it is unanimously impermissible to supplicate for them with mercy or forgiveness.

Supplicating for a living disbeliever with mercy or forgiveness is understood to mean seeking their guidance to Islam, so that God may grant them success in returning from polytheism to monotheism, thereby making them worthy of forgiveness and mercy.

In summary: It is better to supplicate for your mother with guidance and success. If you supplicate for her with mercy, let it be with the intention that God grants her the guidance by which she becomes deserving of mercy.