Does the prayer of one who adds "Our Lord, to You be all praise and thanks" after rising from rukūʿ, or "My Lord, forgive me and my parents" between the two prostrations, or "I seek Your help" in Al-Fatiḥah, or glorifies when encountering a verse of glorification, become invalidated, whether that is in a fard (obligatory) or nafl (supererogatory) prayer? And does the invalidation of his prayer differ if he is convinced of its impermissibility or not?

All the additions and phrases mentioned in the question do not invalidate the prayer, whether done intentionally or unintentionally, because they are of the same genre as the speech legislated within the prayer, not human speech. The ruling on adding the word (wal-shukr - and gratitude) when rising from bowing, the ruling on adding (wa li-walidayya - and for my parents) after the supplication for forgiveness between the two prostrations, and the ruling on saying (ista'antu bika ya Rabb - I seek Your help, O Lord) during the recitation of Al-Fatiha, have been clarified in Fatwa no. 99752.

As for glorifying Allah (tasbeeh) at a verse of glorification, supplicating (du'a) at a verse of supplication, and seeking refuge (ta'awwudh) at a verse of seeking refuge; these are established from the actions of the Prophet, peace and blessings be upon him. Scholars differed on whether this is specific to supererogatory (nafl) prayers or general to both obligatory (fard) and supererogatory prayers.

Regarding restricting oneself to saying "Rabbana laka al-shukr" (Our Lord, to You be gratitude) without mentioning al-hamd (praise), the Hanbalis consider the prayer of one who intentionally does so, knowing the ruling, to be invalid, while the majority of scholars believe the prayer is not invalidated. And Allah knows best.