The results of a new study show [1]:

"Contrary to previous research, caffeine may not relieve movement symptoms for people with Parkinson's disease".

The research paper is behind a paywall, but the abstract says [2]:

"Caffeine did not provide clinically important improvement of motor manifestations of PD (Class I evidence). Epidemiologic links between caffeine and lower PD risk do not appear to be explained by symptomatic effects."

Reference:

[1] That cup of coffee may not relieve Parkinson's symptoms -- ScienceDaily

[2] Caffeine as symptomatic treatment for Parkinson disease (Cafe-PD)

John