I think you need to see the therapist and see what they say before stressing too much. Some counselors are excellent but most counselors can’t prescribe and student health center may only prescribe certain things and so if you need medication you may have to see psychiatrist - if not on staff at school then you’re back to parents seeing EOB for visit. Check into what your school has, but first step is to see therapist and get initial evaluation.
Also, in initial post you say that your doctor is referring you to see someone while at school - do your parents know about this much of the story? Wouldn’t they already know you have some issues?