Prince was known for having had a difficult childhood: His parents divorced before he was ten, and he was on the move constantly as he stayed in different homes and changed addresses more than 30 times as he grew up. In a 1981 Rolling Stone interview, Prince is described as having "pretty much raised himself from the age of twelve."
As for his specific relationship with his parents, Prince’s dad, John Lewis Nelson, was a jazz musician. Even though Nelson named Prince after his band, Prince Rogers Trio, because he “wanted [Prince] to do everything I wanted to do,” growing up Nelson did not provide the support or encouragement Prince needed to become a musician. Prince told Tavis Smiley in a PBS interview that his dad wouldn’t let him play the piano as a child because he was “not as good as him.” According to Touré's biography, "I Would Die 4 U: Why Prince Became an Icon," their relationship improved towards the end of Nelson's life.
As for Prince’s mom, in his four-decade long career, you can practically count the number of times he’s ever mentioned her with one hand. Eric Leeds, brother of Prince’s former tour manager, said in "I Would Die 4 U" that “he spoke of his mom very, very, very rarely. That was a subject that you never wanted to bring up to Prince...It was not a secret that he harbored a lot of resentment toward his mother.” Everyone in Prince’s camp that Touré interviewed agreed: Prince’s mom was a sensitive topic that was to always be avoided.
How Prince felt about each parent is best represented by their characters in "Purple Rain."
Every major character in the film plays themselves, with the glaring exception of Prince’s parents. Prince plays himself, but is named The Kid because the film’s director saw Prince as a “vulnerable kid.” The Kid’s mother is the only major character that does not have a name. She is simply credited as mother. The Kid’s father is credited as father but has a name and a central character of The Kid’s growth. (The Kid’s mother is also white in the film, and Prince’s mom is black.)
Whether all of that was done consciously or subconsciously, it’s a pretty accurate representation of Prince’s relationship with each parent.