Oh I can relate to the embarrassment!!!! I once sent my therapist a text message detailing all of the violent sexual things I wanted to do to him.

Once I discovered it, I really was soooooooooooooooooooo embarrassed to ever set foot in his office again. Let alone look him in the eye.
It's ok that your other parts do things that you find embarrassing, so long as it's only to your therapist. It will mean that your therapist can help that part work through why they feel like that. It will also mean that your therapist will be accepting of you regardless of the embarrassing thing you did, and that feels AMAZING. Then once you realize your therapist will be accepting, it is easier to let the other parts say the things they need to say to heal and you will not feel so embarrassed.
So I'd suggest, don't try to stop the other parts from saying embarrassing things to your therapist. Instead try to talk over with your therapist why it is that you feel embarrassed about what the other part said and talk to your therapist about how it made them feel to receive such a message. I think you will find it turns out for the better.