Perhaps they are blackmailed. A powerful dark wizard could threaten to harm all of their friends and family if they don't abandon Harry. They may feel they have no choice but to sacrifice their relationship with their son to protect others they care about.
In some fanfics, it could be due to a threat she believes she can't protect Harry from. Maybe she thinks that by leaving him, the dark forces won't target him as much.
One reason could be jealousy. Snape might be jealous of James' popularity and charm. James, on the other hand, could be jealous of Snape's close friendship with Lily in their early years. Another reason is their different personalities. James was outgoing and bold, while Snape was more introverted and brooding. Lily might be attracted to different aspects of each of them, which could lead to the fight as they both compete for her attention.
One reason could be that he's tired of the constant danger and politics in the magical world. He might want a normal life away from all the dark wizards and power struggles.
In the fanfiction, perhaps differences in their future plans might lead to divorce. Maybe James wants to move to another city for a job opportunity, but Lily doesn't want to leave her family and friends behind. So, this fundamental difference can drive them apart.
One possible reason could be an alternate universe scenario. Maybe in that fanfic's universe, there are dark forces at play that make Lily and James act out of character and ignore Harry. It could also be a plot device to create a more angsty or dramatic story, where Harry has to face challenges without the support of his parents.
Well, in some versions of the 'Dursleys abandon Harry fanfiction', they abandon him due to social pressure. They are worried about what their neighbors or friends might think if they find out about Harry's true identity. Also, Petunia may have some deep - seated jealousy towards her sister Lily for being a witch, and this negative feeling extends to Harry, leading them to abandon him.
Perhaps in the fanfiction, James is seen as more accepting of Lily's friends and her social circle. Snape, on the other hand, might be associated with a darker crowd or have some beliefs that Lily doesn't agree with. For example, if Snape is too involved with the Dark Arts in the fanfiction, it could be a turn - off for Lily.
Perhaps in this fanfic, Harry has been influenced by dark magic or some evil entity. This external influence could distort his perception of Lily and turn his normal feelings for her into hatred. Dark magic has the power to corrupt one's mind and change relationships in unexpected ways.
It could be an exploration of the darker side of human nature. Writers may use these characters to show how relationships can go horribly wrong, even if it's not in line with the original story.
Maybe some fanfiction authors just like to explore extremely dark and unexpected scenarios. They want to break the traditional image of the characters to create a unique and controversial story.