I’ve heard this one sometimes called ‘alternate biography’. You take someone you know from your life, someone who you know the ‘back story’ of well, and change it.
But what I want us to do is look at all the root causes of what changed. How did that world differ to this? What major events did/didn’t happen and how did that affect their life? Could something so small as a landmark she used to walk by on the way to school wasn’t there in this other world, thus meaning she walked a different path to school so met a different guy to your father, who failed to become the sweet first love story and sent her on a horrid downward spiral?
Or maybe genetic engineering was available back then thanks to advanced aliens so your mother looked completely different but somehow events still conspired to lead her to your father?
And what twists can ou throw in? Could your real mother and your alternate mother somehow meet? Or does your father or you bump into her?
Don’t feel like you have to focus on your parents only either, these were just the ideas my brain jumped to. Where does your creativity take you? Feel free to post links to any resulting stories in the comments.