I do not think a case can be made for calling the Parable of the Prodigal Son the Parable of the Prodigal Father. In the parable both the son and the father are prodigal. They both spent their possessions recklessly, the son "wasted his substance with riotous living", and the father, "he divided unto them his living", which I don't think is very economical and quite in abundance. The father is also prodigal towards his son when he returns broke and in shambles by putting on him the best robe, a ring on his finger, and shoes on his feet, then he tops it off by killing the "fatted calf" for a feast in celebration for his return. .
Also, besides being extravagant with his offerings to his son, both before he left and after he came back, the father is prodigal with his attitude towards his youngest son. The father gives the impression that he did not care the slightest about the fact that his son went off and blew his possession the he had given him. Not only that but he saw him with "compassion, and ran, and fell on his neck, and kissed him." After the son confessed that he had sinned against him and heaven the father praised him even more by clothing him in fine wear and feeding him with feast. .
Although the father is prodigal and maybe even more so than his son, I do no think the story could be called The Parable of the Prodigal Father because there is no moral. Parables are supposed to be stories which a lesson is drawn and I see no lesson in this story by that name. Nothing about the father's excessive giving or pacifist attitude towards his son gives a lesson that should be taken to heart. A more fitting name would be The Fable of the Prodigal Father because then it could be read and the reader would not have to think or worry about whether they are getting or understanding the meaning of the story or not.