What are the responsibilities of the offspring towards their deceased fathers or mothers?

Like in life, parents are entitled to certain rights after their death. Abu Usayd’s narration outlined these rights when he said: “We were with the Messenger of God when a man asked, ‘O Messenger of God! Is there any act of dutifulness which I can do for my parents after their death?’ The Prophet (peace and blessings be upon him) replied, ‘Yes, there are four things you can do: supplicate for them, ask God the Almighty to forgive them, fulfill their pledges, and be generous to their friends. You only have ties of kinship through your parents’” [recorded in Al-Adab Al-Mufrad].
There are five rights which we can list in brief below as follows:

• Pray for them and ask God the Almighty to grant them His mercy and blessings. God the Almighty says: “…and say, ‘Lord, be merciful to them both, as they raised me up when I was little’” [17: 24].
• Ask God the Almighty to forgive them as expressed in the above verse.
• Fulfill their pledges (their wills and testaments) after their death for God the Almighty says: “… after any bequest he [may have] made or debt…” [4: 11]. It is also incumbent to settle their outstanding debts.
• Be dutiful to their relatives, visit their friends and be charitable to them. The Prophet (peace and blessings be upon him) visited and maintained ties with the friends of his first wife Lady Khadijah. Abdullah Ibn Umar said: “I heard the Prophet (peace and blessings be upon him) say, ‘The most beneficent form of excellence is being dutiful to the friends of one’s parents.”
• Be hospitable and generous to their friends by all possible means. Abu Burda is reported to have said: “I came to Madinah and Abdullah Ibn Umar came to me and said, ‘Do you know why I came to you?’ I said, ‘No!’ So he said, ‘I heard the Prophet of God (peace and blessings be upon him) say, ‘He who wants to maintain ties with his father in his grave, let him maintain ties with his father’s friends.’ My father Umar and your father maintained brotherly love for each other so I wanted to maintain that.’ ”