One possible reason could be a misunderstanding. Maybe Naruto gets involved in some dangerous situation and they disagree on how to handle it. For example, if Naruto wants to take on a powerful enemy alone but Minato and Kushina think it's too risky.
Another reason might be related to the village's politics. If there were some decisions being made in Konoha that would affect Naruto's future, like whether he should be sent on a dangerous long - term mission. Minato, as a former Hokage, might have one view and Kushina, as Naruto's mother, might oppose it strongly enough to come to blows.
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.
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.
Perhaps it has to do with a prophecy. They might think that Naruto has a special destiny and that their leaving is a necessary step to set that destiny in motion. They could believe that Naruto needs to grow up on his own to become strong enough to face future challenges.
Perhaps in this fanfiction, there is some sort of misinformation. For example, someone has spread false rumors about Naruto being a threat to the village, and Minato and Kushina believe it without proper investigation. This could lead to their hatred.
One possible reason could be a misunderstanding. Maybe some false information was spread about his parents that made Naruto believe they were bad. For example, if there were rumors that Minato sacrificed the village for some other purpose rather than to save it, Naruto might be angry at him.
Well, it could be related to Minato's absences. He was often away on important ninja business. Naruto might have felt abandoned by his father, and Kushina, seeing her son suffer in Minato's absence, could start to resent Minato. Also, there might have been some political decisions Minato made that didn't sit well with Kushina. For example, if he made an alliance that she thought was against the values of the village or their family.
It could be that Naruto is trying to prove himself. He wants to show that he can be as great a fighter as his father. Also, he might have received some training or knowledge transfer from someone who knew Minato well. This could have inspired him to adopt Minato's fighting style. He may also be in a situation where his normal fighting style isn't working, so he turns to Minato's as a last resort.
If the fight got really intense and out of hand, it could also make Naruto feel pressured to choose sides. However, Naruto being Naruto, he would most likely try to find a way to reconcile them instead. This would show his growth as a person and his ability to rise above family disputes.
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.
The family dynamic in fanfiction could be quite interesting. Minato could be a strict but fair father, pushing Naruto to be the best ninja he can be. Kushina would balance this with her motherly love, always there to comfort Naruto when he fails. They would celebrate Uzumaki traditions together, and Naruto would learn about his parents' pasts. He would look up to Minato as a great Hokage and to Kushina as a strong - willed Uzumaki. Their family would be a source of strength for Naruto as he grows up in the ninja world.