Is adding the phrase "for me and my parents" when supplicating between the two prostrations—which is: "O my Lord, forgive me, have mercy on me, grant me well-being, provide for me, and guide me"—considered an innovation (bid'ah)? And if it is not, what is the difference between this and not adding the word (and gratitude) when rising from bowing?

The worshiper should not persistently add "and for my parents" in the supplication between the two prostrations or "and gratitude" in his praise after bowing, as this has not been authentically reported from the Prophet, peace and blessings be upon him. Persistently doing so would make it seem like a prescribed Sunnah (definition). It is permissible to supplicate for parents during prostration and after the Tashahhud (definition). What is prescribed is "O Lord, forgive me," or "O Allah, forgive me, have mercy on me, grant me well-being, guide me, and provide for me," or "and mend my affairs."