This is a really interesting and important question. I have worked for the same large & well known UK company since diagnosis 11 years ago. I don't have experience of moving to a different company though I have moved roles within the same company a couple of times.
On balance my inclination would be not to mention parkinson's on your cv, on any application form or at interview. As I understand it an employer is not allowed even to ask you about medical conditions and/or disabilities. I think this covers helshubby's point about being 100% truthful.
On the other hand if you need adjustments to the job (location, hours of work, assistive software etc) then these can only be provided after you have informed your employer. My experience is pretty much in line with yours & Sheffy's i.e. that most people are supportive.
As you say your position is different as you are presumably subcontracting a service to a client so you can build your requirements & preferences into the contract. I think this means that even more than an employee you are free to pick and choose who you tell and how you do it.
I recommend the booklet that Sharon mentioned above. It is focussed on direct employment but it covers the main principles and has some good links to more information
I hope that helps