So there is a phase 1 clinical trial in New York but only 8 participants will be enrolled. Clinicaltrials.gov shows this study as recruiting but I have to wonder why such a small study is still recruiting. I suppose it would be worthwhile to contact the researchers if a person happens to live near this site (NYU Langone Health). So if you page down into the study they generally have the contact information.
https://www.clinicaltrials.gov/study/NCT0563487...

This is the Vaxxinity site and it shows the projects they are working on and what stage they are in. The Alzheimer's research is advancing to phase 2. The Parkinson's research has not yet advanced to phase 2 and phase 3 clinical trials but at this point it is not possible to find out where these will take place.
https://vaxxinity.com/our-pipeline/

Our Pipeline - Vaxxinity

https://vaxxinity.com/our-pipeline/

I have been trying to find out if there will be further clinical trials on this and so far not finding studies other than a very small study in New York state that is not for Parkinson's but is for multiple system atrophy
https://www.clinicaltrials.gov/study/NCT0563487...

sorry this the correct link https://time.com/6883054/parkinsons-test-vaccin...

A Potential Parkinson's Treatment Has Promising Results

https://time.com/6883054/parkinsons-test-vaccine-study-vaxxinity/