What are some ways to deal with choking and dry mouth when suffering from parkinson's disease?

Nasal obstruction: Dry mouth or "cotton mouth" usually indicates possible nasal obstruction that promotes mouth breathing. See an ENT phsycian to rule this aspect out. It can be serious enough to progress to sleep disturbances like obstructive sleep apnea that can be life shortening or life threatening. I encourage you to check it out.

Mouth Moisturizers: Many of the drugs used to treat parkinson's disease cause decreased saliva and dry mouth. There a number of commercial "artificial salivas" that vary in effectiveness from patient to patient, so if one is not helpful, try another brand. Biotene makes a whole line of oral care products for dry mouth suffers, which many of my patients find soothing, regardless of the cause of dryness.