If both 120 and 240V appliances could be wired with either parallel or tandem plugs at one time, it seems like it could have been a disaster waiting to happen for someone.
This continues to cause confusion to the present generation of electricians.
The parallel vs. tandem issue should be considered separate from the 125 vs. 250 issue. In the early days, the medium Edison base was the only universal wiring device standard in the USA (and only after 1900, as the Thomson-Houston and Sawyer-Mann bases were also common before that). Several companies in the first two decades of the century produced two-piece "cap and base" plugs in different competing configurations. The most common of these were the parallel and the tandem. By 1936 (actually probably by the late '20s) the parallel device was most common, yet there were still enough tandems around to make T-slots a good idea.
The voltage issue is quite another matter. Several Electrical Nostalgia posts question the reason for a 10A 250V rating. Prior to about 1920 (I'm not exactly sure when, or which antique book I read this in, although they are all on my shelf), lighting circuits were to be fused at 10A. Originally, I think they were limited to 6A, as the medium Edison base was very early on limited to 660W. After about 1920, lighting circuits were allowed to be fused at 15A, as that is the actual ampacity of 14 AWG (2.08mm2) rubber wire. I guess the 250V rating was frozen at 10A since these devices were probably never used much at 250V.
I should mention that in those days there was no really strict rule limiting a particular receptacle configuration to a particular voltage. I gather that 32VDC farm installations used the exact same wiring devices as 120VAC installations at the time (I know for certain they used medium Edison-base devices). I've also heard that 120VDC installations in major cities used the same receptacles as AC installations elsewhere.
I have a variety of old devices here in a box:
Hubbell tandem Edison plug base, rated 300V 3A.
"C-H" (Cutler-Hammer?) "loop-prong" cap and base rated 660W 250V.
GE T-slot receptacles, one a single and the other a duplex, rated 10A 250V.
Porcelain Paulding T-slot duplex receptacle, rated 15A 125V and 10A 250V.
Arrow-Hart duplex T-slot (which I removed from the 1890s house where I lived in college), 15A 125V and 10A 250V.
I think that the tandem configuration was thought of as just another 120V configuration.