After quite a bit of searching, I was unable to find any conclusive authoritative source that weighs in specifically on this question. [Though see EDIT below.] However, circumstantial evidence suggests that Turing certainly popularized the term "imitation game" (whether or not he coined it himself). Furthermore, I've found no evidence of a popular parlor game resembling Turing's description, played in the Victorian era or otherwise. That doesn't mean it didn't exist, but what seems clear from publications around the time of Turing's original paper is that if Turing's supposed parlor game existed already, no one seemed to reference it or understand what Turing's reference was to. Hence, if Turing didn't make it up himself for his example, it likely was a rather obscure game, rather than a "popular Victorian game" as is sometimes claimed.
To provide some further details, let's first address the term "imitation game" itself. Aside from the Google Ngrams data in the question that suggests the term's popularity increased significantly after Turing's paper, there are several sources in the years immediately following Turing's paper that imply that the term "Imitation Game" was of Turing's invention. For example, note the wording of Leonard Pinsky in the very first academic reply to Turing's paper "Do Machines Think about Machines Thinking" (1951):
The new formulation [of the question whether machines can think] was
based upon what Mr. Turing called the "imitation game".
Follow-up papers similarly refer to the test as "Turing's 'Imitation Game'" (1958) or "Turing's well-known 'Imitation Game'" (1959). Note the quotation marks in each case, likely indicating a novel term, rather than a previously known one. From this (and other contemporary examples) it seems likely that Turing popularized the term "imitation game" and this specific use of the phrase. If the phrase existed earlier, it appears Turing's usage was novel at a minimum, leading to the quotation marks in early citations. As further evidence, I'd note that early references like this that don't specifically credit Turing will still say something like "Turing describes an imitation game" or "a game of sorts." I haven't come upon a single reference that claims there was a specific pre-existing known "imitation game" (i.e., the Imitation Game) that Turing might have been referring to.
As for the possibility of a pre-existing parlor game, I came upon several modern references that are more specific than the BBC link in the question. Perhaps the most intriguing is a 2014 Smithsonian Magazine article claiming:
[Turing] took the idea from a traditional Victorian parlor game, where you
try to figure out if the person hidden behind a curtain is a man or a
woman, just by asking questions. (The answers to the questions had to
be written down, because the voice would be a giveaway.)
That's much more detailed than most descriptions, mentioning the idea of a "curtain" as well as the parameter that "the questions had to be written down" (though from a practical standpoint, it would have been the answers rather than the questions that would have to be written down). This article, like all the other recent ones that assume the existence of the Victorian parlor game, provides no citation or clear reference detail.
Nevertheless, with these details in hand, I skimmed through descriptions of at least a couple hundred different Victorian parlor games in various collections of them published in the 1800s, and I haven't seen anything mentioned that resembles this game. One of the only popular games involving a "curtain" was about casting shadows on the curtain (or sheet) and requiring a person behind the curtain to guess who was making the shadows.
Other recent sources claim that the people had to be taken to a different room, which actually makes more sense, as visual and auditory cues would likely make it much easier to guess who was who simply behind a curtain or sheet in the same room. Nevertheless, while there are many Victorian parlor games involving one or two people leaving the room, they inevitably tend to quickly return -- usually having concocted some sort of scheme or idea that the rest of the group has to guess. After all, the goal of Victorian parlor games was socialization, and a game that banished a few members of the party to another room for a long period of writing responses wouldn't be particularly social.
In any case, having reviewed several popular books that each collected many dozens of common games, I'm ready to say this game -- if it existed in Victorian times -- likely wasn't a well-known or very popular game. (And there was certainly no game known commonly as the "imitation game.")
There's also a lot of circumstantial evidence that these modern assumptions of a "Victorian game" are referencing a myth. Careful writers frequently avoid attributing a specific reference to a pre-existing game. The Smithsonian quotation above is one of the few I've seen in a reputable source that seems to provide any details at all. Others often hedge their bets, like the BBC article quoted in the question, which only claims that the competition was "Victorian-style", not an actual Victorian game. And the BBC lets open the possible implication that Turing himself came up with this "adaptation" of a game he might have himself invented.
That latter implication is also made by several other careful writers. Daniel Dennett says Turing "described a parlor game of sorts, the 'imitation game', to be played by a man, a woman, and a judge (of either gender)." Note the reference to a "parlor game of sorts," which seems to argue against a pre-existing game. Similarly, a 2019 New York Review of Books article describes Turing's scenario as follows:
Turing’s imitation game, as he set it out in 1950, has two stages.
In the first, we measure how well humans can distinguish between a
woman and a man who is impersonating a woman. Then we see whether
humans are better or worse at telling the difference between a woman
and a computer imitating a woman.
As Turing described it, three people would take part in the first
stage of the game. [Description of the man, woman, and judge scenario
follows.] Now, asked Turing, “What will happen when a machine takes
the part of A in this game?” In the second stage, the man in the
second room is replaced by a computer program.
There's no mention of any pre-existing "parlor game" here, Victorian or otherwise. The article simply assumes there are two "stages" to the game, all apparently attributed to Turing himself. Given how much discussion there has been about gender issues in this game, it's also rather surprising that I haven't been able to find a single such source that connects this back to an actual parlor game. (See, for example, recent articles here and here: the first merely noting that Turing "suggested an 'imitation game,' which plays like 20 Questions for transsexuals" and the second simply calling this game a "scenario." Neither makes any reference to a "parlor" or Victorians.)
Indeed, in this entire 2004 book on the Turing Test, with numerous reprinted historical essays and references to various controversies around it, there's no mention of the "parlor game" interpretation. In fact, the only time the term "parlor game" is even used in the entire book is in Daniel Dennett's essay (referenced above), where it's clear that Dennett doesn't think this is a pre-existing game, just a "parlor game of sorts."
Nevertheless, Turing's original game does resemble other actual Victorian games in some ways. There are plenty of Victorian parlor games involving someone hiding a secret, or requiring someone to guess the identity that another person is trying to portray, or games where people try to deceive others by disguising their identity in some fashion. (Usually the games involve blindfolding the person guessing.) Hence, the BBC could still accurately describe Turing's scenario as a "Victorian-style competition." In fact, even contemporary references around the time of Turing's paper make comparisons to Victorian-era games, such as W. Mays's 1952 article "Can Machines Think?", which describes Turing's scenario like this:
[Turing] substitutes a definition in terms of the machine's capacity
to play an imitation game which seems to be a variation of the radio
game of Twenty Questions, the part of the witness being taken by a
computing machine, the part of the interrogator by a human being.
Twenty Questions was a radio show at the time, but it also had a long history going back to Victorian parlor games. Clearly, even to Turing's contemporaries, his "imitation game" sounded something like a parlor game to them, just not like any precise game they'd ever had heard of (or else they'd reference that actual game, rather than something like Twenty Questions). Nevertheless, despite many recent references to a supposed popular Victorian game (maybe involving a curtain and other details), I couldn't find any evidence of its existence prior to Turing's imaginings. If it previously existed, it clearly wasn't very well-known.
EDIT: I did manage to find a pretty authoritative academic source that appears to weigh in rather conclusively. Mark Bishop, a professor of cognitive computing at the University of London, began an issue of the journal Kybernetes devoted to the Turing Test in 2010 with an introduction called "The Imitation Game" by noting:
Famously, in his 1950 paper ‘Computing Machinery and Intelligence’ ,
the British mathematician Alan Turing suggested replacing this
question ["Can a Machine Think?] - which he found “too meaningless to
deserve discussion” - with a simple [behaviourial] test based on an
imagined Victorian-esque pastime he entitled the ‘imitation game’.
Bishop minces no words here: the "imitation game" was named by Turing, and it was only an imagined Victorian-esque pastime. Given this claim in a prominent place in an issue of an academic journal devoted to discussion of the Turing Test, I'd guess that Bishop would have been pretty sure of himself before writing these words. Nonetheless, he's not the only academic to say something about this topic, and there are contradictory examples, such as Barry Luokkala (from the Department of Physics at Carnegie-Mellon) who writes in Exploring Science Through Science Fiction (2019, p. 130):
The [Turing] test was based on the Imitation Game, a parlor game
popular in the pre-television era.
But Luokkala, like other similar sources, provides no citation or evidence supporting this claim. In the absence of anyone anywhere providing any citation of this supposedly "popular" game, and many academics like Bishop assuming it was of Turing's invention, I'm personally willing to declare the pre-existing "popular Victorian game" to be a myth.