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.
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.
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.
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.
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.
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.
One possible reason could be some sort of dark force influencing Kushina's mind. Maybe she was under a powerful genjutsu that made her think she had to abandon Naruto for his own safety, even though it was a false perception.
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.
Perhaps in this fanfiction, Naruto turns to the darker side of the ninja world. Since he lacks the love and guidance from Minato and Kushina, he might be more easily influenced by Orochimaru or other villains. He could start to believe that power is the only way to get respect and recognition, and thus embark on a path that is different from his canon self. He might also form unexpected alliances with characters who are also outcasts in the shinobi society.
Well, it could be that they find out some 'dark secret' about Naruto. Maybe they think he has some sort of uncontrollable power within him that they fear. Or perhaps they are forced to disown him due to political pressure in the village. There could be a group of elders or powerful clans that demand it in order to protect their own interests or maintain the so - called 'balance' in the village.