If an adult goes to counseling and reveals that she was abused as a child, will the parents be turned in by the counselor?

Unlikely: If childhood abuse isn't disclosed until adulthood it is very unlikely that a counselor would turn in the parents. It becomes a he said she said situation at that point because there would probably no evidence that the abuse occurred.

No: Many adults who were abused or neglected as children, seek out psychotherapy. Their parents are not "turned in" by therapists or psychiatrists, but the adult patient is worked with as needed. The important thing is to assist the patient in front of us, to recover and heal. The only time we might have to report, is if there's a child still in the home who might be harmed.