I don't think this is necessarily the case.
My dad had Parkinsons. My mom, who is 79, has started doing the same thing, her head shakes slightly when watching the telly. But it is nothing like the tremors my dad had in the slightest. There are other symptoms to Parkisons that are likely to come first too. Does he drag his feet when walking? Is he lacking in co-ordination?
I think that by inisisting someone goes to the doctors, you're likely to worry them. I personally think it's more just associated with age and nothing to worry about. But then again, I'm not a doctor - you have to make that decision. I certainly won't say anything to my mom, because I think she'll feel her age more. If she feels unwell, she'll tell me.
Hope that helps?