There might be a very dark and twisted fanfiction like that. It could be that Naruto is tricked into believing that Minato and Kushina are some kind of threat to the village or to his friends. So in a moment of confusion and misinformation, he takes drastic action. But this goes completely against Naruto's character in the original series.
One possible plotline could be that Minato and Kushina train Naruto to master his chakra control more effectively. They might use their own unique jutsu and skills to teach him. For example, Minato could teach Naruto advanced space - time ninjutsu techniques that he himself was famous for, like the Flying Thunder God Technique. Kushina, on the other hand, could focus on helping Naruto understand the power of the Nine - Tails better and how to use its chakra in a more harmonious way. This could involve special training sessions where Naruto has to learn to balance his own chakra with that of the Nine - Tails under their guidance.
One possible plotline could be that Minato and Kushina focus on teaching Naruto advanced chakra control. Since Minato was known for his Flying Thunder God Technique which required precise chakra control, he could pass on those skills. Kushina, with her strong chakra, might teach Naruto how to better harness his own Uzumaki chakra reserves. For example, they could start with basic exercises like tree - walking and water - walking, but with a twist. Minato could show Naruto how to use his chakra in a more efficient and strategic way during these exercises, preparing him for more complex jutsu later.
Well, if there is a 'naruto kills kakashi fanfiction', it would be a very different take on the Naruto universe. It could be set in a world where Naruto has been corrupted by an evil force. This force could be manipulating Naruto's emotions and memories, making him think Kakashi is a threat. In the original series, they have a strong bond, so this would be a very controversial and unexpected plot.
There could be a 'naruto kills jiraiya fanfiction'. It could be based on a time - travel scenario. For example, if Naruto from the future travels back in time and somehow his actions cause a chain of events that lead to Jiraiya's death. Or perhaps there's a dark, alternate universe where Naruto's values are twisted and he sees Jiraiya as an obstacle to his new, malevolent goals and thus kills him.
A possible plotline is that Naruto gets adopted by a group outside the village. He grows up learning their ways and becomes a sort of outcast from Konoha. Then, he could return to the village with a different set of skills and a new perspective. Or, he might develop a vendetta against Konoha for allowing his abandonment and try to take revenge in some way.
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.
One possible plotline could be that Kushina is somehow mind - controlled or influenced by an evil force to harm Naruto. Maybe she's under a powerful genjutsu that makes her see Naruto as a threat to the village instead of her son.
On a more personal level, they could face internal guilt. As Naruto grows up and achieves great things on his own despite their neglect, Minato and Kushina may realize what they've missed out on. This realization could cause them a great deal of internal turmoil and regret for not being there for their son when he needed them.
The implications would be huge. It would completely shatter the positive family image that the Naruto series usually portrays. Fans who love the relationship between Naruto and his parents would be very shocked and possibly offended.
One possible motive could be that Minato is under some sort of dark magic or mind control. In fanfiction, authors often play with the idea of characters being forced to act against their true nature. So, if Minato is being controlled, he might end up doing something as extreme as killing Naruto.