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.
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.
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.
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.
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.
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 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.
Running away can make Naruto more independent. He has to learn to fend for himself, which would strengthen his survival skills. He would also become more self - reliant, not relying on his parents' support or resources.
Emotionally, Naruto would have to deal with a whole new set of family relationships. He grew up without his parents, and now having them back would be a huge adjustment. He might struggle to fit into a 'normal' family dynamic, and there could be some friction as they all get to know each other again.
In some fanfictions, Naruto might find a super - powerful jutsu or artifact. For example, he could discover a long - lost forbidden jutsu that has the power to bring the dead back to life. Or perhaps he stumbles upon a magical item during one of his adventures that is said to be able to revive the deceased.