Would there be any advantage to doing this instead of just having parallel single-ended tubes?
Not that I know of.
My thought is: in class A, there’s no efficiency gained by running push-pull instead of parallel-single-ended, but the push pull arrangement lets me squeeze a little more power out of two tubes by giving the option of class AB?
When a tube is run single-ended (and not as you're thinking, with 2 output transformers), then the tube distorts when driven to:
1) Plate-current cut-off, or
2) 0v grid-to-cathode (which results in grid-current).
To get the biggest output, the tube is typically biased about halfway between these 2 points (if plate current cuts off at Grid at -35v, idle grid-to-cathode voltage might be -15v).
When tubes are run push-pull, plate current cut-off on one side may not result in distortion if the other side is still conducting. It depends on whether the tubes are being run in "Class A," "limiting Class A," "a little into Class AB" or "deep Class AB."
Read Radiotron Designer's Handbook 4th Edition for an explanation of "limiting Class A," which basically means one side is cutting off just as the other side reaches grid-current. Overall, Class A means peak plate current on one side is 2x that tube's idle plate current.
Whether the output tubes are running Class A, "a little into Class AB" or "deep Class AB" is decided by the overall balanced of supply voltage (including screen voltage), OT primary impedance, and idle bias.
Cathode-bias push-pull tubes can run "Class A" or "a little into Class AB" depending on the factors noted above. "Class AB" mostly means peak plate current is higher than "2x idle current" (sometimes very much higher); obtaining the high peak plate current requires higher supply voltages, lower OT primary impedance, and fixed-bias.
The “downside” of the push pull circuit (traditionally) is the need for a phase inverter, though I do often enjoy the sounds produced by not-quite-symmetric phase inverters.
100 years ago, creating a phase inverter might have been a challenging task. Today, it's pretty trivial.