Yes, it is possible in fanfiction. Fanfiction writers often explore 'what - if' scenarios. They could imagine a scenario where Batman has some sort of precognitive ability that he didn't know about before. With this ability, he could foresee the attack on his parents and stop it. This would create a whole new narrative for Batman, where his origin story is completely rewritten and his relationships with other characters in the Batman universe would also be different.
Well, the 'found family' trope can be used in a different way in this type of fanfiction. Instead of Batman creating a new family with Alfred and his allies, he gets to keep his original family. There's also the 'hero's journey reversed' trope. Usually, Batman's journey starts with the loss of his parents, but in these fanfictions, that loss is averted, and we see how his character develops in a different direction. And the trope of 'unexpected ally' can be used, like a villain who helps Batman save his parents for some self - serving reason.
In Naruto fanfiction where Naruto saves his parents, one way could be through time - travel. Maybe he discovers a jutsu that allows him to go back in time. He could then use his knowledge of future events and his already - developed skills to prevent their deaths. For example, he might be able to stop the Nine - Tails attack on the village that led to his parents' sacrifice.
Darkseid is another possibility. He's a powerful and menacing villain. If he sets his sights on Wonder Woman, Batman would be forced to act. He'd need to use all of his skills and resources to save her from Darkseid's Omega Beams and his army of minions.
One storyline could be that Jason discovers a plot against Batman early on. He then sets up a counter - plan and at the crucial moment, jumps in to save Batman from a deadly trap. This would show Jason's intelligence and his lingering care for Batman despite their past issues.
Well, it might be that in the fanfiction, there's a parallel universe where Batman's parents didn't die. So he meets them in this alternate reality. He has to adjust to a world where he doesn't need to be the brooding vigilante. Instead, he gets to experience a normal family life, but he also realizes the importance of his role in his own universe and takes some valuable lessons back with him.
Maybe Snape has some sort of premonition or knowledge that Harry is going to be crucial in the fight against dark forces in the future. So, in order to ensure that future, he steps in to save Harry from whatever danger his parents' situation might put him in. He might also be trying to atone for past mistakes or grudges he held against Harry's father by protecting his son.
One possible scenario in Batman saves Batgirl fanfiction could be that Batgirl is in a dangerous situation, like being held captive by a powerful villain. Batman, with his detective skills and combat abilities, manages to find her location and battles the villain to rescue her. It's a classic superhero - damsel in distress (although Batgirl is also a superhero in her own right) situation that often plays out in fanfiction.
Obviously, Batman himself would be one of the main characters. And of course, his parents, Thomas and Martha Wayne. There could also be Alfred, as he is an important figure in Batman's life and might play a role in facilitating the meeting or providing some insights afterwards.
One of my favorite scenarios in Batman saves Catwoman fanfiction is when it takes place during a city - wide disaster. Batman has to fight through chaos to find Catwoman, who might be trapped in a collapsing building. It shows his determination to save her no matter what.
There are also fanfictions that explore the aftermath of the rescue. How does Catwoman react? Does she become more indebted to Batman or does it create some sort of power shift in their relationship? These stories can really dig deep into the complex relationship between the two characters.