Maybe in the fanfiction, Harry's parents are alive but held captive by a dark wizard. The Dursleys take Harry in as a way to keep him safe from the same fate. Harry might grow up with a sense of mission to rescue his parents from a young age, and the Dursleys, despite their usual nastiness, might have a glimmer of kindness knowing that Harry's parents are heroes. This could create a complex web of relationships and emotions within the story.
One possibility could be that in this fanfiction, Harry's parents are in some sort of magical stasis or hidden away. The Dursleys, perhaps forced by some magical contract or under a spell, reluctantly take Harry in while his parents' situation is being sorted out. This could lead to a very different dynamic than the canon, with Harry constantly longing to find his parents and the Dursleys being more on edge as they have to deal with a magical child while his powerful parents are still in the picture.
The plot could be that Harry's parents are in a magically-induced coma. The Dursleys are tasked with looking after Harry until a cure can be found. Harry might feel a sense of isolation as he is with the Dursleys but constantly thinking about his parents. He could start to show early signs of magic, which terrifies the Dursleys even more, but also gives him hope that he can somehow use his powers to wake his parents up.
Another reason could be that they had some sort of premonition or prophecy that indicated the Dursleys' place was a crucial part of Harry's future protection. Maybe a seer told them that for Harry to grow into the wizard he needs to be, he has to endure the Dursley experience first.
In some fanfictions, it might be due to an alternate reality. For example, if there was a different prophecy or a different turn of events at the time Voldemort attacked. Harry could leave his parents to fulfill a new - found destiny in this altered world. Also, it could be because of a miscommunication. A magical misunderstanding might lead Harry to believe he has to leave his parents for their own safety.
One possible plotline could be that Naruto is banished due to false accusations of endangering the village, despite his parents being alive. His parents, Minato and Kushina, might be powerless to stop the decision because of some political intrigue within the village. Naruto then has to find his own way in the outside world, facing new enemies and making unexpected allies.
Well, in fanfiction, it might be that a magical object has the power to bring the dead back to life. For example, a long - lost and extremely powerful enchanted amulet. When Harry discovers this amulet, it has the ability to restore his parents to life. It could also be that a benevolent and extremely powerful being decides to intervene and resurrect the Potters for the sake of Harry and the greater good of the wizarding world.
If Harry is abandoned by his alive parents in fanfiction, it could completely change the story. For one, he might not grow up with the Dursleys. He could end up in a different magical or muggle family. This new family could either be very loving or have their own ulterior motives. His personality might also be different as he wouldn't have the same experiences of being an unwanted child in the Dursley's house. He could be more confident or perhaps more bitter depending on how his new family treated him.
It can make him more independent. Without his parents' protection, Harry has to learn to fend for himself, which can lead to him becoming more self - reliant. He'll have to figure out how to solve problems on his own.
Well, for one, Harry would grow up in a loving family environment. He wouldn't have to live with the Dursleys. His magical education would likely start much earlier and be guided by his parents, especially James who was also a very talented wizard. His confidence and self - esteem would be much higher as he wouldn't have the emotional baggage of being an orphan.
The plot would change a great deal. For one, Harry wouldn't have to live with the Dursleys. He would grow up in a loving magical household. He'd be more confident and perhaps less of an underdog. His parents would be able to train him in magic from a young age, so he might be more advanced in his studies when he gets to Hogwarts. Also, the fight against Voldemort would be a family affair, with James and Lily Potter being key players in the battle against the Dark Lord.
In such a fanfiction, Harry would likely be in shock. His world view has been centered around their deaths for so long. He could be hesitant to trust this new reality. But once he does, he'd have a whole new support system. His parents could teach him magic they've learned in their 'hidden' years. This would make him even more powerful and give him a different standing among his peers.