Interesting. Neurogenesis I think yes, very possible, but I don't see how one can do enough exercise in PD patients (60-80 yr) in a way that can increase neurogenesis in a significant way.

Here's an article featuring the potential benefit of exercise in treating Parkinson's: http://giving.med.ubc.ca/a-surge-toward-new-solutions-for-people-with-parkinsons/

Here is a review of studies of patients as well as animal models investigating the underlying mechanism of exercise-induced neuroplasticity in PD - while the clinical literature is still limited, it does seem like a promising line of research in the management of the disease:

Article Exercise-enhanced Neuroplasticity Targeting Motor and Cognit...

Neurogenesis of DA neurons in the substantia nigra region in an inflammatory response model of Parkinson's disease was reported in this study: Article Discovery of nigral dopaminergic neurogenesis in adult mice

Whilst one would expect the same mechanism to work in principle across species, I am not aware of any clinically meaningful effect proven in humans. Also, one should bear in mind that disease models, help us to understand certain aspects of human conditions, but rarely grasp the full picture.

Still, physical activity has many additional positive effects beyond inciting neurogenesis and it is certainly a good thing to encourage patients to do within their individual limitations.

I guess one of the mechanisms which contribute to the positive outcome of exercise therapy for Parkinson's patients could be increased cerebral bloodflow: Article Regulation of Cerebral Blood Flow During Exercise