I read a “fact vs fiction” article about the movie, there was an interview where Elton said he just wanted his father to notice his accomplishments and he never did. And apparently what little relationship they had was done after Elton talked about him very unflatteringly in an interview in the ‘70s. Although apparently he did visit with him and his half-brothers on various occasions, not just the one time.
As for his mother, they went a long time without talking and then he talked to her shortly before her death.
So not great relationships.
Though the movie makes out his mother as not wanting to hear anything about him being gay, while IRL apparently a point of contention was his mother’s continuing friendship with his ex John Reid after everything that went down.