One idea could be exploring how his parents' abilities were passed down to him at a genetic level even as a baby. Maybe baby Naruto shows early signs of the Nine - Tails' chakra influence in a cute yet powerful way, like his eyes glowing red when he's really excited or angry.
It could be that in a special mission, Naruto discovers a hidden jutsu that allows him to communicate with the spirits of his deceased parents, leading to a touching reunion.
The relationship between Naruto and his parents can also be portrayed with a sense of destiny. Since Naruto is the host of the Nine - Tails, his parents' sacrifice is always in the background. In some fanfics, they might leave behind hidden messages or powers for Naruto that he discovers as he grows up, strengthening the bond between them even in their absence.
Perhaps Naruto has always felt different from others in the village. And when he finally learns about his parents, he realizes that their actions, even if they were for a good cause like protecting the village, still made his life very difficult. He might be angry that they didn't find a better way to handle things so that he didn't have to suffer so much growing up.
His relationship with his parents would be loving. Minato would be a strict but fair father, teaching Naruto about ninjutsu and the responsibilities of being a shinobi. Kushina would be the warm, affectionate mother, always there to comfort and encourage Naruto. They would support his dreams and be proud of his every achievement.
It might make him more distant from his friends at first. He could be so consumed by his anger towards his parents that he withdraws from his usual social circle. For example, he might avoid spending time with Sasuke or Sakura because he is too preoccupied with his feelings about his parents.
A plot for this type of fanfic could revolve around Naruto's inner turmoil. He could be constantly haunted by the question of why his parents neglected him. This could lead him to make some bad decisions, like getting involved with the wrong people. However, through a series of events, he starts to realize that he can create his own family and support system. He then begins to build a new life for himself, one that is not defined by his parents' neglect but by his own strength and the relationships he chooses to cultivate.
In this kind of situation, Naruto might first feel extremely sad and lonely. He could start by trying to prove himself to others. Maybe he'd focus on his ninja training more than ever, using it as a way to distract himself from the pain of being forgotten by his parents.
The village would be in chaos. Naruto was seen as the hope for the future, so his death would leave a huge void. His friends like Sasuke and Sakura would be devastated. They might turn against Naruto's parents, leading to more conflict in the village. Also, the balance of power in the ninja world could shift as Konoha would be weakened without Naruto.