One interesting fanfic could be where Harry is abandoned as a baby but is found by a group of magical creatures who raise him. They teach him unique magic that Hogwarts doesn't offer. Another might be a story where Harry is abandoned and ends up in a different magical realm altogether, having to find his way back to the wizarding world while uncovering secrets about his parents.
Often, it changes the power dynamics. Instead of growing up with the Dursleys, Harry might grow up in a completely different environment, which gives him different skills and knowledge. For example, if he's raised by magical creatures, he could have a deeper understanding of nature magic.
You can start by looking on some of the major fanfic websites like Fanfiction.net or Archive of Our Own. Use the search function and type in 'Harry Potter' along with 'abandoned' to filter out the relevant stories.
Maybe they were forced into it by some dark magic or a powerful dark wizard. They could have been blackmailed into betraying Harry to save their own lives or the lives of others they cared about.
A fanfic could revolve around a Hufflepuff whose parents are alive and own a magical business. The Hufflepuff student has to balance school life with helping their parents out during holidays. This could lead to all sorts of adventures as they meet different magical creatures and characters through the business. At the same time, they use the Hufflepuff traits of kindness and patience to deal with difficult customers and situations.
One possible fanfic could be about a wizard who discovers an ancient and forbidden magical ritual. He decides to use it to change his parents' personalities. Maybe he wants them to be more lenient or more adventurous. He secretly gathers the rare ingredients and performs the ritual at midnight in an abandoned castle. However, things go wrong and instead of the expected changes, his parents start acting completely out of character, like his mother suddenly having a passion for Quidditch and his father becoming obsessed with knitting.
One possible fanfic could be about Harry's parents being magically changed in appearance. Maybe they were cursed by a dark wizard and had to live with different looks. This could add a whole new layer of mystery and drama to the story, as Harry might not recognize them at first if they were to somehow be brought back to life in this changed state.
Well, there are indeed. The concept of Harry being neglected by his parents in fanfics gives a whole new perspective. It allows the author to explore how Harry's prodigious abilities might manifest in a more desperate situation. He might be more resourceful and independent, which is a fresh take on the Harry Potter character. Some fanfics also show how this affects his relationships with other characters in Hogwarts.
A third plotline might be that Harry is abandoned in a magical but hidden place. There, he is raised by magical creatures, say, a group of centaurs. They teach him their own brand of magic and wisdom. When he finally enters the wizarding world proper, he has a very different perspective on magic compared to the other students, which causes both admiration and some envy among his peers.
One possible plotline could be Harry growing up in an orphanage where he discovers some latent magic powers on his own before getting his Hogwarts letter. He might face more difficulties and prejudices from the muggles around him, which could make his entry into the wizarding world even more of a relief.