Perhaps it has to do with a prophecy. They might think that Naruto has a special destiny and that their leaving is a necessary step to set that destiny in motion. They could believe that Naruto needs to grow up on his own to become strong enough to face future challenges.
One possible reason could be some sort of dark influence or curse that makes her lose her maternal instincts. Maybe a powerful enemy targets her and in order to protect Naruto from further danger related to her, she feels forced to abandon him. Another reason might be that she is made to believe false information about Naruto, such as that he will bring great disaster to the village in the future.
It might be due to jealousy. If Minato was getting more attention or praise in the village for his achievements while Kushina felt overlooked, that could create resentment. Say, Minato was constantly being hailed as a hero for his speed and battle tactics, and Kushina, who also had great power, felt like she was in his shadow.
If Minato and Kushina abandon Naruto in a fanfic, it could lead to a much darker Naruto. He might grow up more jaded and bitter. Without their love and support, he could be more easily influenced by negative forces. Maybe he'd have a harder time believing in friendship and loyalty, which are such important themes in the original story.
One possible reason could be a misunderstanding. Maybe Naruto gets involved in some dangerous situation and they disagree on how to handle it. For example, if Naruto wants to take on a powerful enemy alone but Minato and Kushina think it's too risky.
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.
The story could also show Naruto being taken in by other characters, like Iruka at a very early age. Iruka, seeing Naruto's situation, decides to be like a father figure to him. Naruto might then develop a different set of values based on Iruka's teachings rather than the absent influence of Minato and Kushina. He could become more focused on family values within the village, seeing the village as his true family since his biological family left him.
Well, it could be that they find out some 'dark secret' about Naruto. Maybe they think he has some sort of uncontrollable power within him that they fear. Or perhaps they are forced to disown him due to political pressure in the village. There could be a group of elders or powerful clans that demand it in order to protect their own interests or maintain the so - called 'balance' in the village.
Naruto could be found by Orochimaru. This would be a very dark turn. Orochimaru could use Naruto's potential for his own evil plans. Naruto would be trained in Orochimaru's unethical ways, and his view of the ninja world would be completely distorted. He might become a villain or at least an anti - hero, very different from the Naruto we know in the original story.
In some cases, Naruto might initially become bitter towards his parents. But as he experiences more in life and learns about the possible reasons behind their actions, he could come to understand and even forgive them. His reaction would also depend on the people around him. If he has good friends or mentors, they could help him deal with his emotions in a more positive way.
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.