Paying off a debt of one’s parent after their death:
A child is not obligated to pay off his father’s debt, unless his father has passed away and he has inherited money or assets from his father, in which case he is to use the inheritance money, or assets, to pay off his father’s debts. However, if his father did not leave any money, then there is not even a Mitzvah involved in using his own money to pay it off. This applies even if his father instructed him to pay off his debts. Certainly, one is not obligated to pay off his father’s debts using his own money while his father is alive, even if his father instructs him to do so.
 Admur C.M. Mechira Halacha 7; Michaber C.M. 107:1; Hagahos Ashri Kesubos 9:14; Smeh 107; Gilyon Maharsha 240; Pesakim Uteshuvos 240:32; Encyclopedia Talmudit ibid footnote 1168-1184
Other opinions: Some Poskim rule that due to the mitzvah of honoring one’s parents, the children are obligated to pay off their father’s debt even if they did not inherit any money from him. [Hagahos Ashri ibid in name of Riy; Maharam 76; Rashba on Kesubos 86a; Igros Moshe Y.D. 2:63; Encyclopedia Talmudit ibid footnote 1181]
 Michaber ibid; However, see Pesakim Ueteshuvos ibid footnote 273 who explains that nonetheless a son who does pay off the debt is certainly doing a good thing for the soul of their parent
 Admur ibid