If Naruto is pushed away by his parents in the fanfic, his character development would take a very different turn. He might be more introverted initially, hiding his pain. His view of family would be distorted. But as he discovers friendship, like with Sasuke and Sakura, those relationships would become even more crucial to him. He'd learn to create his own family unit in a sense, with his friends filling the void left by his parents.
Maybe he discovers some dark secrets about his family. For example, if his parents were involved in something he considered unethical in the fanfic's plot. It could be that they were part of a group with a hidden and evil agenda.
Another possibility is that they were deceived. There could be a villain who tricked them into believing that giving away Naruto's birthright was the right thing to do. They might have been misinformed about the consequences and the true nature of what they were giving up.
Well, he could meet new friends or allies who help him grow stronger. For instance, he might encounter a group of outcasts who understand his situation and teach him new skills or ways of thinking.
One possible reason could be that he discovers a dark secret about his family. Maybe he finds out that his parents were involved in something he can't accept, like a deal with a dangerous enemy. So, to avoid being associated with it, he runs away.
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.
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.