Maybe in this fanfic, there has been a misinformation spread about Naruto that makes Kushina believe he has done something unforgivable. For example, if false rumors say that Naruto is involved in something evil, Kushina might start to hate him without knowing the truth.
Maybe in this fanfiction, there was some sort of dark magic or mind - control involved. Kushina could have been unknowingly influenced to do something that deeply hurt Naruto. Like she was made to abandon Naruto at a crucial moment in his life, which then led to his hatred.
One possible reason could be a misinterpretation of Kushina's actions. Maybe some events were presented in a way that made Naruto think she was neglectful or had made bad decisions in the past.
One possible reason could be that Dark Naruto might see Kushina as a symbol of the world that rejected him. Since she was an important part of Naruto's family and the village, and Dark Naruto represents Naruto's negative emotions, he could resent her for the life he didn't have. He may feel that if Kushina had been there for him more (in his twisted view), he wouldn't have turned into this dark version.
One possible reason could be a misunderstanding. Maybe some false information was spread about Kushina that made Naruto believe she was a bad person. For example, if there were rumors that she abandoned him for some untrue reason, it might lead to his hatred.
One possible reason could be a misunderstanding. Maybe some false information was spread about his parents that made Naruto believe they were bad. For example, if there were rumors that Minato sacrificed the village for some other purpose rather than to save it, Naruto might be angry at him.
Maybe in this fanfic, the author wants to create a unique and dark plot. Kushina might be made to hate Naruto due to some kind of misunderstanding. For example, she could be misled to believe that Naruto has some connection to a great evil that she is against.
Perhaps in this fanfiction, there is some sort of misinformation. For example, someone has spread false rumors about Naruto being a threat to the village, and Minato and Kushina believe it without proper investigation. This could lead to their hatred.
One possible reason could be a misunderstanding. Maybe some false information was spread about Kushina that made Naruto believe she had wronged him in some way. And for his sister, perhaps there was some sort of jealousy or miscommunication. For example, if his sister got more attention from their father or something similar.
Maybe she was forced to for some dark reason. Perhaps there was a threat to Naruto's life if she stayed, and she thought leaving was the only way to protect him.
It could also be due to a misunderstanding. Kakashi might be keeping some secrets from Naruto, like information about his parents or some hidden power within him. Naruto, being curious and eager for answers, could misinterpret Kakashi's motives for secrecy as a sign of distrust or dislike, which then turns into hatred.