The story would change a great deal. For one, Naruto wouldn't have the same connection to the Nine - Tails as he does in the original. Minato sealed the Nine - Tails into Naruto, so without that relation, the whole dynamic of the Nine - Tails within Naruto would be different. He might not have the same power - up moments related to the Nine - Tails.
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.
Well, in such a fanfic, Naruto could be the son of someone like Jiraiya. Jiraiya was very close to Naruto and taught him a great deal. He might have had a secret relationship that led to Naruto's birth. Another possibility could be that Naruto is the son of an unknown shinobi from a far - off village who somehow ended up in Konoha.
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.
One scenario could be that Naruto discovers some hidden secrets about Minato and Kushina that he interprets as them being selfish. For instance, he might find out that they had a choice to be with him but chose something else instead, which makes him angry and full of hatred towards them.
One possible way this fanfic could go is that Minato and Kushina's spirits could be in a state of limbo and they are forced to witness Naruto's abuse. This could lead to a series of events where they try to find a way to communicate with Naruto or influence the situation to stop the abuse. Maybe they reach out to other characters in Naruto's life to help protect him.
One consequence could be that Naruto becomes an outcast in a different way. In the original, he was an outcast because of the Nine - Tails, but here, it could be because of his abandonment. He might not have the same dreams of becoming Hokage. Instead, he could be more focused on survival.