How do I find a mental health therapist in my area (Nashville, Tennessee) that has experience with Parkinson’s patients and their caregivers?

I would call Licensed Social Workers listed as in private therapy practice. They are more likely to have a grounding in this. You might go to support groups FIRST as you may find guidance there for folks in your area with some grounding. Only other way to find out is to ask, and you can ask online with Parkinson's Groups as well. Good luck in your search.