I think if Palpatine had introduced any part of himself into Rey, then he almost certainly would know or have a say in where she was going to be born. Therefore, it wouldn’t make sense for her parents to be involved with the Sith unless it was expressly to return Rey to the Sith. At least with the way we’ve framed things now, Palpatine is only claiming creations of the Force as his rightful “property”. This doesn’t give us the implication that he got to decide where the child went to, and open up that plot hole of him not immediately coming for a young Rey.

I honestly think that giving Rey a further portion of the dark side is unnecessary. We’ve positioned things in quite a clever way already for there to be double meanings that Rey initially takes to mean the worst:

“Palpatine influenced the Force itself into creating life”. Rey and Kylo take this to mean they are rightfully Palpatine’s, destined for darkness like Vader.
Luke tells Rey that this is a stretch of the truth. The Force only made them to destroy him.
Rey confronts Palpatine armed with this knowledge, and he turns the tables once more: “Yes, I know you are meant to destroy me. That’s what I want and intended all along for you. My spirit will pass into you, and you will become a Palpatine regardless.”
It’s only because of the Force ghosts that she can fulfill her purpose ordained by the Force without succumbing to Palpatine’s machinations. She claims herself a Jedi, and later a Skywalker since both her own parents and Palpatine never had any right to her.