Yes, Barry, you are exactly right. I see it all the time on American records. Immigrants would often Anglicize their parents names on records whether or not the parents immigrated. Having said that, make sure that the parents didn't come at a later date if you have confirmed that they didn't immigrate with the child or children.
I have been surprised several times at finding parents or just a widowed parent immigrating in their later years. I say that it is surprising because the family stories for those lines indicated that the children immigrated but the parents stayed behind. And that wasn't the case. The parent or parents came later to join their children.
Risa Daitzman Heywood