Given that handwriting is a forgotten art and a lot of individuals have nearly indecipherable handwriting (and the inability to spell) these days, are you sure anyone would notice if you didn't bring it up?
It's really an issue for an AME, rather than a designated examiner.
With Parkinson's, a tremor is often only perceptible under certain conditions; one may be fully dexterous, yet have a repetitive tremor while making a motion to screw in a lightbulb, for example.
I wouldn't let your instructor or anyone else make a medical diagnosis. Leave that to a doctor. Keep in mind that the majority of flight instructors have no experience; they've never worked in aviation or done anything but instruct, and are brand new pilots themselves. With that in mind, they may be the last ones you want to turn to for information about the industry, or what is or is not acceptable.