Ozy & Millie was one of the first webcomics I ever started reading regularly, and I still rate it as one of the best webcomics ever produced.
But no, the "parcheesi" joke predates even O&M by a long ways. It was first associated as a "metaphor for sex" in a 1964 Audrey Hepburn movie, something like "Paris when it Sizzles", so something like that.
She plays a secretary to a lazy writer, and at one point she's reading a partial script, and expresses annoyance when she comes to the part where the scrips goes "... and the couple leans closer, kisses, and '...'".
She asks why always use that kind of literary subterfuge, since the reader always knows the couple is going to have sex.
The writer counters that she must have a dirty mind, since for all you know, they could just be playing Parcheesi".
It doesn't hurt that it's a funny-sounding word, either. It wouldn't sound as flippant if he'd said "playing chess" or "playing dominoes". O&M almost assuredly used it for the same reason- the name itself has a funny tone.