Sorry to hear about your Greyhound. Typically, no. Dogs do not get Parkinson's but they are prone to other neurological diseases that can cause head shakes and tremors. These can include things like:
1. Canine vestibular disease- This typically happens in older dogs and shows up with a head tilt, head shakes, stumbling gait, and eyes that shift back and forth. It's due to an issue with a nerve that runs through the inner ear and in older dogs it can spontaneously resolve (go away on its own). Here's a website with more info:
2. Central brain lesions (typically in the cerebellum) which can be due to several diseases including neoplasia (brain tumor- can be non-cancerous or cancerous), vascular accident (like a stroke or a blood clot), congenital anomaly (birth defect....unlikely due to the age of your dog), a tick born disease (may or may not be an issue in your area), bacterial diseases (like toxoplasmosis or neosporum), viral diseases (like distemper...again unlikely due to the age of your dog) and fungal disease.
I know I've thrown a lot at you here, but the best place to start is with a vet visit. They can evaluate your dog and give you some more information as to which of these they think may be occurring. If they feel it is vestibular disease, they may ask you to wait it out for a week or two as occasionally this disease will resolve on its own. If they feel that it is likely a cerebellar issue they will probably want to refer you to a neurologist to better evaluate your dog. The neurologist may recommend doing some blood tests, doing a spinal tap, doing some imaging (like a CT scan or MRI) or may just be able to help differentiate the possibilities based on the examination alone.
I hope this information has been helpful and good luck with your dog!