Perhaps in the fanfiction, Harry's parents are actually trapped in a sort of limbo. They manage to break free and come to Harry's aid just when he is at his lowest point. They might have gained special powers in the limbo which they use to save Harry. For instance, they could be able to freeze time momentarily and remove Harry from a dangerous situation. Then, they start to train Harry in magic that is not taught at Hogwarts, magic that was passed down through their family but was lost when they died.
One plot could be that Harry's parents are brought back to life through some dark or ancient magic. They return to the wizarding world just in time to save Harry from a powerful dark wizard. Their return also has a big impact on the wizarding community. People start to question the nature of life and death. Harry's parents then help him uncover some secrets about his family that were buried long ago, which in turn gives Harry more strength and knowledge to face his future challenges.
This is a very dark and unorthodox concept. In a possible plot, perhaps an evil force could be manipulating Harry's mind, making him believe his parents are a threat to the wizarding world. But this goes completely against the essence of the original story where Harry loves and respects his parents.
One possible plot could be that Harry grows up in a loving and magical home. He gets to learn magic from his parents at a very early age. His father, James, might teach him Quidditch skills right in their backyard. His mother, Lily, could pass on her knowledge of potions and herbology. They would also protect him from the dark forces that are still out there, perhaps leading to some epic battles where the family stands united against the remnants of Voldemort's followers.
Another possibility is a group of Death Eaters who have broken away from Voldemort's control and formed their own faction. They see Harry as a symbol of the old regime that they want to overthrow. They target Harry, but his parents, with their knowledge of the old ways and their love for Harry, are able to save him from this group's attacks.
The plot would change drastically. Harry would not be living with the Dursleys, so there would be no need for him to escape that horrible situation. Voldemort might have a harder time targeting him as his parents would be on high alert. There would be no letter - hunting drama. Also, his relationship with Snape would be different as his father was Snape's rival. His parents could have taught him some advanced magic from a young age, which would make him stand out at Hogwarts in a different way than in the original story.
In such a fanfiction, perhaps the parents were forced to betray Harry due to some dark magic or a powerful enemy threatening them. Maybe they had to give up Harry to save the wizarding world from an even greater evil.
It could be that baby Harry's parents take him on a secret trip to a hidden magical place. They want to keep it a secret from the wizarding community for some reason. During this trip, they teach Harry about the importance of family magic and the history of their family's unique magical abilities. This journey strengthens their bond as a family and also gives Harry a head start in understanding his magical heritage.
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.
Harry would probably be overjoyed at first. After all, he has always longed for his parents. He might start to look for ways to make it happen, like searching for magic spells or ancient artifacts that could bring his parents back.
He would probably be very confused and heartbroken at first. He might isolate himself from others as he tries to make sense of what has happened.
In a fanfiction where Harry Potter reads his parents' will, it could lead to him uncovering secrets about his family. Maybe he'd find out about hidden magical items or undisclosed family alliances. This could change his perspective on his parents and his place in the wizarding world.