I only know PD and not long COVID, but it was news to me that there is a skin biopsy test for PD. Neurologists told me PD was determined by observation, and in my case, assisted by a PET scan. Some people have mentioned a DAT scan too, but here is a page that seems pretty official:
https://www.mayoclinic.org/diseases-conditions/parkinsons-disease/diagnosis-treatment/drc-20376062
I am also not sure what the usual PD tremor is. My point is that I am not saying this is PD and I have no idea what would cause such fatigue (I am not doctor, just someone who has had early onset PD for the last 15 years.) Five years is a long time, but I do know that my PD makes every other malady worse. I also have to treat my asthma, and exercise also helps me feel better. Usually though, PD makes you sleep less, not more. Sorry for your situation.