It does sound like Laryngopharyngeal reflux (LPR) for all of you above! It's really uncomfortable, but also fixable.
Your GP can help. A good treatment is a combination of Proton Pump Inhibitor (PPI) medication, Gaviscon Advance and lifestyle changes. This can be really successful, but you need medical input, particularly for taking other meds with Parkinson's meds.
There's more information on reflux on my website (southbristolvoicetherapy.co.uk) and if you email me I can send you a useful indicator for LPR to take to your GP.