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.
One possible reason could be for Naruto's protection. His parents might think that by giving away his birthright, they can keep him away from dangerous enemies or political intrigue in the ninja world. Maybe they foresee a great threat that would target Naruto specifically because of his birthright, and this is their extreme measure to ensure his survival.
This change in the fanfic would likely have a huge impact on Naruto's character. Without his birthright, he may be more distrustful of the system that allowed such a thing to happen. His usual optimism might be tinged with cynicism. For example, he might be less likely to believe in the Hokage system if it was involved in taking away his birthright. He'd have to re - evaluate his goals and dreams, and this would be a major part of his new character development.
He might be initially very confused and hurt. Naruto has always longed for a connection to his parents, and finding out they gave away his birthright could make him feel like he was unwanted or that they didn't trust him with it. He could go through a period of self - doubt, wondering if he is really worthy of being their son.
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 Danzo. He was always power - hungry and had his own ideas about what was best for the village. He might see Naruto as a threat and find a way to take away his birthright.
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.
In such a fanfiction, perhaps Naruto could be wrongly accused of something that leads to him being stripped of his birthright. Maybe a false claim about his connection to the Nine - Tails is used against him.
One possible reason could be that they are too involved in their own shinobi duties. The world of ninjas is full of dangerous missions, and they might be constantly away on important tasks, leaving Naruto feeling neglected.
In this fanfiction, it could be that Naruto's parents were under some sort of mind - control. Some evil entity could have taken over their minds and made them feel hatred towards Naruto. Also, it might be related to a family secret or feud. There could be something in the family's past that made them believe Naruto was a stain on their family name, even though Naruto himself was innocent. It could also be a case of false information being spread around the village which made his parents believe lies about him.
Maybe they were under some kind of curse or dark magic that made them fear they would harm Naruto if they were close to him.