Bert Kalmar and Harry Ruby wrote dozens of satirical or downright silly songs during the golden era of Tin-Pan Alley's heyday. I do several of their tunes, including "Father's Day" a song they wrote for Groucho Marx (they wrote several for him, including "Lydia the Tattooed Lady") and "The Sheik of Avenue B", a parody of "The Sheik of Araby" that they wrote for Fanny Brice.
This tune is clearly a mild parody of Madame Butterfly, but it's a thin little piece of fluff that doesn't need to be analyzed, just played.
A quick thought on the subject and it's treatment, though. It has a mildly racist approach, but that's typical of what was a far more unapologetically racist time, the 1920's. And when you come down to it, this tune isn't even remotely malicious, so I wanted to do it. However, I don't know that I'd feel comfortable playing "Ma is Playing Mah-Jong".