There is nothing wrong in telling the truth. It is not a secret. Your parents know it is true.
Nobody knows who you really are or cares who you are,… at least I don’t.
If you think it was a malious gossip type of talking behind someone’s back then it is a problem. Especially if you feel guilty.
What your parents don’t know will not hurt them. I prefer people not tell me what other people say behind my back because it really hurts my feelings. It is best to keep silent if you feel you made an error. None of us are perfect.