It could be a matter of prophecy. Minato might have received some sort of prophecy that staying with Naruto and Kushina would lead to the destruction of the village. So, in a misguided attempt to save Konoha, he abandons them. Or, there could be a power struggle within the village that makes Minato believe that leaving Naruto and Kushina is the only way to keep the peace. However, these are all just speculations within the context of fanfiction.
In such a fanfiction scenario, the village of Konoha would be affected too. Minato is a respected figure. His abandonment could cause a scandal. Naruto might not have the same level of protection from the village elders. This could lead to different power dynamics within the village. Maybe other ninja clans would try to take advantage of Naruto's situation. Also, Kushina's family might react in various ways, some might try to find Minato and some might focus on taking care of Kushina and Naruto.
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.
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.
In many Naruto fanfictions where Naruto meets Kushina and Minato, it often creates a very emotional scenario. Naruto finally gets to meet his parents who he has only heard about. It could be a heart - warming reunion where they share stories, hugs, and Minato and Kushina get to see how far Naruto has come.
Minato would probably be very proud. He might notice Naruto's use of his own jutsu and be impressed by how Naruto has developed his skills. Kushina would be more emotional, maybe crying at the sight of Naruto growing up so well.
Well, in some fanfictions, the moment when Minato sacrifices himself to save the village while Kushina is already dead is extremely emotional. It shows his selflessness and also the love he has for the village and his family. Also, the flashbacks in some stories about their early days together, like when they trained together as young ninjas and slowly fell in love, are really sweet and emotional.
The family dynamic in fanfiction could be quite interesting. Minato could be a strict but fair father, pushing Naruto to be the best ninja he can be. Kushina would balance this with her motherly love, always there to comfort Naruto when he fails. They would celebrate Uzumaki traditions together, and Naruto would learn about his parents' pasts. He would look up to Minato as a great Hokage and to Kushina as a strong - willed Uzumaki. Their family would be a source of strength for Naruto as he grows up in the ninja world.
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.
Perhaps in this fanfiction, Minato and Kushina leave Naruto because they are cursed. They fear that the curse will harm Naruto if they stay close. So, they go into exile. As Naruto grows up, he starts to discover clues about his parents' real reason for leaving and sets out on a quest to find them and break the curse. This could lead to many adventures and character - development moments for Naruto.
One possibility is that Naruto has an out - of - body experience and is able to communicate with his parents in a spiritual realm. They could give him advice on how to deal with his enemies and his inner demons.