Fanfiction about Naruto being abandoned by Minato and Kushina can also be a way to explore different power - up scenarios. Since he doesn't have the emotional support from his parents, he might be forced to seek power in other ways, which could lead to a Naruto with a very different set of skills and abilities compared to the original series.
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.
There's a fanfic called 'Minato's Return'. In this story, Minato somehow returns and gets to see Naruto's growth. It's really interesting to see how he interacts with Kushina, who is also there in the story, and how they both guide Naruto in new ways. The descriptions of their family bond are heartwarming.
One common theme is isolation. Naruto would be all alone without his parents, and this would shape his relationships. Another theme is revenge. In some fanfictions, he might seek revenge on the village for his abandonment.
Well, in such a fanfic, Naruto would probably have a very different personality. He might not be as optimistic. His training might take a different path too. Instead of having the Nine - Tails as a sort of legacy from his father, he could see it as just a curse that was left with him when he was abandoned. His bond with the village might be weaker as he was rejected by the two most important people who were supposed to be from the village in the first place.
There would be a lot of anger. Naruto might be angry at his parents for leaving him, and this anger could fuel his determination in some ways. For example, he could strive to become a great ninja just to prove that he didn't need them. But it could also cause problems in his relationships, as he might lash out when reminded of his abandonment.
One great fanfic is 'The Early Days of Naruto' which beautifully depicts Minato and Kushina's relationship with baby Naruto. It shows their tender moments together and how they start to teach Naruto about the ninja world from a very young age. Another good one is 'Naruto's Family Beginnings' where the focus is more on the family dynamic. You can really feel the love Minato and Kushina have for their baby. And then there's 'Baby Naruto's World' which goes into detail about how Naruto's presence affects the village even as a baby, and how Minato and Kushina are at the center of protecting and nurturing him.
In some fanfics, Naruto might be made to hate Minato and Kushina due to a sense of abandonment. If the story is written in a way that emphasizes how Naruto grew up alone and suffering, he could project his anger onto his parents for not being there for him. For example, he might think that if they were alive, he wouldn't have had to endure the taunts and loneliness in the village.
In a fanfic, Kushina might be depicted as a source of great wisdom for Naruto. Since she was his mother, she could pass on some Uzumaki family secrets or just some motherly advice. Minato could be the guiding force in Naruto's development as a ninja. His skills and experiences could be something that Naruto looks up to. Naruto himself should be a complex character, with his own insecurities and doubts at times, but always pushing forward because of his parents' influence.
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.