What if you told the counselor about your parent's mistreatment of you, the lack of support you receive from them over mental illness, and your concerns that it will backfire against you by asking the counselor for help? It couldn't hurt to explain your situation at home, your concerns and fears, and see what the advisor has to say? She/he may have a solution for you. At the very least, it's a step you could take.

As far as internalizing or believing your parents' unjust words towards you, it is not your fault. It is not your fault that you are suffering, or that you have mental illness. None of that is your fault, so please don't believe their false words.

You need to look at them as emotionally unbalanced people. You need to view them differently. They are not emotionally altogether there. Otherwise, they would be 100% supportive, they would trust you at your word, and they would get you the help and treatment you need. Try to view them as dysfunctional, and therefore, what they say, their words towards you, come from a place of malfunction. Once you see them as being apart from normal parents, then their words towards you won't stick as much or have as much weight or value. What YOU think, feel and believe to be real is most important. You need to feel validated and supported in your struggles.