Parkinson's has, among other explanations, been hypothesized as the result of a viral, bacterial or other infection and/or autoimmunity.
About 25 years ago my aunt was preparing for cancer chemotherapy. She had a concurrent diagnosis of lupus. As related to me by family members, some portion of her medical team told her the lupus would go away after the chemotherapy, and it did.
Two explanations seem particularly plausible.
The predominant medical thinking at the time was that chemotherapy deactivated or destroyed that portion of the immune system responsible for the autoimmune condition we call lupus.
Another, that I think received little conventional medical attention at the time, is that this and other "autoimmune conditions" are actually infections the immune system continues to pursue, without victory, but with civilian casualties in the form of damage to tissues caught in the crossfire.
Yours is a defensive concern. You want to prevent worsening of the Parkinson's -- that could arrive as a result of the chemotherapy. It's a good question.
An equally interesting question is offensive. Does chemotherapy sometimes result in the reversal or "cure" of "secondary" autoimmune and/or neurodegenerative conditions?
To those who are reading: Please relate anecdotal, or "formal" evidence, of 1.) Worsening of the secondary condition after chemotherapy, 2.) "Reversal" of the secondary condition through apparent knockout of some portion of the immune system by the chemotherapy, and 3.) Apparent cure of the secondary condition through possible destruction of one or more pathogens.