From Clarissa Dickson Wright's A History of English Food (2012):
[In the 1920s, British] hostesses also started to serve canapés: little cheese Parmentiers, asparagus rolled in thin brown bread and butter, delicate crab patties, and, of course, the ubiquitous vol-au-vents [...] (source)
Antoine-Augustin Parmentier (1737–1813) was a French proponent of the potato.
Googling "cheese Parmentier" just wants to tell me about hachis Parmentier, basically a French version of shepherd's pie, dating back to at least 1898. ("Hachis" here I think is French for "a chopped-up thing," and has no relation to the Spanish word for "hashish," but I'd appreciate some authoritative enlightenment on that point too.)
Hachis Parmentier does contain both cheese and potatoes, and also phonetically sounds like "(ha-) cheese Parmentier"; the only reasons I'm doubting this identification are (1) that I can't find any evidence at all that anyone's ever actually referred to hachis Parmentier as "cheese Parmentier," and (2) little shepherd's pies sound like they'd make terribly messy canapés.
So, what did Clarissa Dickson Wright mean by "little cheese Parmentiers"?