In the fanfiction, Naruto could justify his hatred because he may have grown up seeing other kids with their mothers. Kushina's absence during his formative years could have made him feel unloved and unwanted. He might think that she chose to leave him, not considering how much he would suffer, and thus hatred could develop.
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.
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.
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 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.
The fanfiction could develop this hatred through the introduction of new characters. These characters might feed Naruto false information about Minato and Kushina, making him question their motives and ultimately hate them. For example, a character who has a grudge against the Uzumaki family could manipulate Naruto.
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.
Perhaps Naruto blames Yondaime for his mother's death. In the original story, Kushina died while giving birth to Naruto, and if in this fanfiction there are some hidden motives or actions of Yondaime related to that event, Naruto could hold a grudge. Like if he found out that Yondaime could have saved Kushina but chose not to for some greater good that Naruto doesn't understand or accept.
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.
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.