Before your father goes directly to a neurologist specializing in motion disorder, please suggest that he see his primary care doctor. Resting or static tremor is a primary symptom of aging and therefore also a symptom of Parkinson's disease which mimics accelerated aging. But tremors can also be a result of cardiovascular issues and medication side effects or drug interactions. They can also be a sign of stress.
His primary care doctor should have a record of all medications your father is taking. If he/she doesn't, please ask your parents to compile such a list. This list includes prescription medications, over-the-counter medications and nutritional supplements which he takes on a regular basis.
Another list that would be helpful is one which lists any symptoms he might have been esperiencing recently even though they were ascribed to something else or just ignored. For example loss of sense of smell can be an early indication of PD. Difficulty sleeping. depression, leg drag, change in hand writing can all be early signs of PD.
Diagnosis for Parkinson's disease is not state of the art yet; it is done through observation and patient history usually facilitated by the UPDRS (or the revised version) as well as lab tests to rule out other conditions.
It is good that you recognized this symptom. Early diagnosis is important in all medical conditions and especially so in PD as there are a few treatment options in early diagnosis one of which has been shown to slow progression.
Please read what you can find about treatment options so that you can provide additional printed matter for your parents to consider should the diagnosis indeed be Parkinson's. Your parents can decide with the doctor what course of medication seems best for them at this time.
Your parents are probably aware of the tremor you witnessed but were not be ready to take action. It may be your role to encourage them to make the appointment.