Did you have the cat scanned* for a microchip at the vet? You must do that!
Just asking if the vet has heard about a missing cat isn’t enough due diligence on your part. Why do you assume it was abandoned? You could be taking someone’s pet. I picture two little doe-eyed girls between 4 and 7 crying themselves to sleep every night after praying to Jesus that their kitty is okay. They put signs up all over the neighborhood, and Dad took them door-to-door asking after the cat because the little girls insisted, but no one knows anything. They are heartbroken. Please return their cat so Jesus won’t have to send you to Hell.
Write back after you do this and we can talk about what to do about your parents.
* Not the same thing as a CAT scan, in case you were wondering.