Setting Snakes on Hapless Relatives
I think that a lot of it depends on Rita's relationship with her parents. If she is a "daddy's girl" she might blab to her dad. However, if it is the other way around she would probably keep her mouth shut. (Or as shut as it can be.)
Either way, I agree with cmwinters. She wouldn't be able to completely shut up about it. Of course, I doubt that she would spread the news too much if it in some way could hurt her or her parents. (Should she care for them.) While I doubt that Rita was ever particularly close to either or her parents, the news of her mother cheating on her father might make people wonder if Rita was born outside the marriage and so on. And Rita probably wouldn't like this too much.
However, Rita might still talk about it, just not as much as if it was someone elses parents. Perhaps she would tell her close friends (if Rita ever had close friends, let alone friends) at a sleepover, or something like that. Perhaps if she found some interesting proof about who her mother was cheating with, she might want to show off that she had the brains to figure this out.
That's all from me.