In some fanfics, they might abandon Harry because they are under a dark spell. This spell could make them lose their love for him and force them to leave. Another reason could be that they are blackmailed by some powerful dark forces. These forces may threaten their lives if they don't abandon Harry.
His life could become very difficult. He might have to fend for himself at a very young age. Maybe he has to live on the streets, stealing food just to survive.
In fanfiction, his alive parents could abandon Harry due to a prophecy. They might have been told that Harry was destined for great evil or something terrible was going to happen because of him. So, in a misguided attempt to prevent this, they abandoned him. Or perhaps they were under some kind of dark magic influence that made them believe that Harry was a danger to them or the wizarding world, even though it was all false.
In some fanfictions, his parents might be forced to abandon him due to a powerful enemy threatening their family. They may believe that by leaving Naruto alone, the enemy will not target him immediately, thinking he's just an ordinary orphan.
Yes, there might be. Some fanfics like to subvert the expected negative situation. In these fanfics, even though the start is that Harry's parents hate him, through a series of events such as Harry showing extraordinary courage or kindness, his parents could have a change of heart. For example, he might save them from a great danger, which makes them realize their mistake and start to love him. This kind of plot twist can add more depth and surprise to the fanfic.
In these fanfics, a common theme is growth. Harry has to grow up faster and become more independent because of his parents' hatred. Another theme is the search for identity. Without the positive influence of his parents, he has to define himself. And the theme of resilience is also prominent. Harry has to be strong in the face of his parents' negative feelings and all the difficulties that come with it. He has to keep going and not let their hatred break him.
In certain fanfic scenarios, Harry's parents may have left behind some enchanted objects. These objects could have special powers. When Harry discovers them, they might activate and create a magical portal or something similar that allows him to leave the Dursleys' place. For instance, a locket that when opened, transports Harry to a safe place where his parents' magic has created a haven for him.
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.
It could be due to the saturation of the market. There might be so much Harry fanfic already that it's hard for new works to stand out. Additionally, changes in the way people consume media, such as a shift towards more visual content like streaming shows, could also lead to a decrease in interest in reading Harry fanfic.
One way could be that they appear as ghosts or apparitions. Their presence alone might be enough to spook the Dursleys into releasing Harry. The Dursleys are very much afraid of the supernatural, and seeing Harry's dead parents would surely be a terrifying experience for them.