Although behind a pay wall, I came across a paper [1] which shows correlations between various PD measures. MDS-UPDRS Part III scores and PDQ-8 [2], a PD quality of life questionnaire, have a correlation of 0.47. If you take the square of the correlation, you get the proportion of the variance that can be attributed to a factor. Here we have 0.47 x 0.47 = 0.22. So, less than a quarter of the variance is explained by the Part III score. In fact the situation is probably worse than that: the PDQ-8 is itself a proxy measure of the true progression of the disease.

Reference:

[1] Eur J Neurol. 2014 Mar;21(3):519-24. doi: 10.1111/ene.12349. Epub 2014 Jan 21.
"Relationship between the MDS-UPDRS domains and the health-related quality of life of Parkinson's disease patients."
Martínez-Martín P1, Rodríguez-Blázquez C, Forjaz MJ, Alvarez-Sánchez M, Arakaki T, Bergareche-Yarza A, Chade A, Garretto N, Gershanik O, Kurtis MM, Martínez-Castrillo JC, Mendoza-Rodríguez A, Moore HP, Rodríguez-Violante M, Singer C, Tilley BC, Huang J, Stebbins GT, Goetz CG.
Relationship between the MDS-UPDRS domains and the health-related quality of life of Parkinson's disease patients. - PubMed - NCBI

[2] Parkinson’s Disease Questionnaire

John