I take coconut oil, based on various research, summarised quite nicely here: http://healthimpactnews.com/2014/parkinsons-sufferers-find-relief-and-improved-life-after-starting-coconut-oil/ it's about £6 a tub in supermarkets, or Costco do much bigger tubs which are significantly better value for money. You can cook with it, or add it to dishes (I take it with PIP Mix (next paragraph) and greek yoghurt.)
I also like the research links between inflammation in general, but in particular gut health and Parkinson's, so take PIP Mix from here: http://www.brainhelp.info/why.html - Andrew Carmichael runs this small charity and I have found him very approachable when it comes to questions and even sharing theories. PIPMix has taste issues, but where there's a will ...
I am inclined to accept that there's something in the hypothesis that fats are good for the brain, so I have started following a largely ketogenic diet: http://www.drdeborahmd.com/solutions-parkinsons-disease adding butter to veg has massively improved my enjoyment of what was already a pretty healthy diet. I love walnuts too, rumoured to be good brain food.
All of these are easily countered with contradictory hypotheses so it's a case of reading up and choosing that which you feel most comfortable with - I consider myself my own walking science experiment, as I don't necessarily have the 5 to 15 years medical science wants before committing to one argument or another.
Do what feels good, keep records, and share what you find.