Hi Marly,
I know it's hard, but you can talk to your therapist. I agree, ask her about what she has to tell your parents. Tell her you want to be open with her, but you worry about it not being able to stay between the two of you. And then, talk to her. She may have had other people struggle with the same issue. This is probably not going to be the first time she has heard this. She wont think less of you or like you less. She will be concerned and want to help. She will also be very happy you were honest with her.