I don't think a naturalization record that early, 1883, would have the parents' names on it. Some doc., later, did ask the father's name, but I don't remember if that was one of the naturalization records or a passport application.
By the way there were 3 docs for naturalization - Declaration of Intention, Petition for Naturalization, and the naturalization itself. You had to wait a certain number of years between the Declaration and the Petition. And you had to be in the US a certain number of years before making your Declaration.
edit:
I had almost posted this when Neale1961 posted, So the next two paragraphs might be unnecessary now.
The fact that Wm (1860, Glasgow) made his own declaration in 1883 probably tells you that his father never was naturalized. I'm pretty sure he would have been naturalized on the strength of his father's. So maybe the father had died, or maybe he never bothered to become a US citizen. You didn't have to.
If you have found most or all of the children in 1880, especially if they are scattered, but not the parents, that would make it pretty likely that at least one of the parents had died before then. At least you likely know where - NY City.
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Marriage records in the US have never asked whether the parents were dead or alive. Not any I have seen anyway.
Wm is in the 1902 Montclair NJ City Directory as
Wm Thomson, arch, 87 1/2 Forest
But is not in the 1904 or 1905 directories. I didn't look any later than that. So your hunch that they moved might be right.
