Is it permissible for a Muslim to add an extra rak'ah to each obligatory prayer for his deceased parent, and to perform Hajj on their behalf after their death?

Allah has legislated filial piety towards parents and being good to them through charity, supplication, and all other forms of goodness. However, performing Salah on their behalf is not legislated. Thus, a child should not pray for their parent; rather, they should supplicate for them, give charity on their behalf, perform Hajj and Umrah on their behalf, offer sacrifices on their behalf, and honor their friends and relatives. As for adding a rak'ah or a separate prayer on their behalf, it is not permissible because there is no textual evidence for it. The Prophet, peace and blessings be upon him, was asked about showing kindness to parents after their death, and he said: "Yes, by praying for them, seeking forgiveness for them, fulfilling their pledges after their death, maintaining the ties of kinship which are not maintained except through them, and honoring their friends." And "praying for them" means supplicating for their forgiveness and mercy. "Fulfilling their pledges" means executing their lawful wills, not those that contradict the Sacred Law. "Maintaining the ties of kinship which are not maintained except through them" means being good to the paternal uncles and maternal aunts, and the like. And "honoring their friends" is among the greatest acts of righteousness.