As-salam alaykum brother Ahmad,
This concept is not supported by the Quran.
Arguably, from a Quran’s perspective, once a person has passed away, their account is closed and is with their Lord for accountability. The best one can do is pray for their mercy and for themselves with no guarantees. It is only Gods will that can affect the outcome.
For example, some claim on the basis of an alleged prophetic report that a child’s prayer will benefit the parent (as you have shared). The inference is that the parent has had some hand in parting good guidance to the child and positively affected their upbringing which has resulted in them becoming righteous individuals who in turn pray for their parent. Notwithstanding the fact that only God can guide (28:56), even prophet Noah’s son was of evil conduct and a disbeliever (11:46, 11:42-43). Are we to assume that prophet Noah didn’t do his best to raise his son in the most righteous way?
One can only be held responsible for the deeds that they have done on earth themselves. This concept of ‘reliance’ on others (or accruing endless rewards after death) to save them in the hereafter is a misguided thought and quite embedded in Muslim thought and other religious factions finding little or no support in the Quran.
017.013
"And We have made every human's actions (augury) (to be) fastened to his own neck, and We will bring forth to him on the resurrection day a book / record which he will find wide open"
It is important to remember that earthly relationships will be disowned on the Day of Judgment where a parent will attempt to ransom their own child, their spouse and the nearest of kin that sheltered them on earth so that they may be saved (70:11-16); where a nursing mother will forget her nursling and where a pregnant mother will drop her load / child in her stomach and will want nothing to do with it (22:2) .
I hope that helps, God willing
Joseph