The premise was widely used in fiction of the time. Works like Who Goes There? and, most famously Invasion of the Body Snatchers, especially popular in the 1950's, expressed the fear that people are not what they seem to be. Dick's story is typically more personal because it is not about the invasion of a community, but of a family.
The Father-Thing is the US Underwood-Miller (1987) and UK title of the third collected volume of Dick's short stories (retitled Second Variety after Second Variety was moved from Volume 2 by Citadel) .