Multiple LiftMaster 882LMW
The short answer to the question is yes, two LiftMaster 882LMW door openers can be wired in parallel and work.

Please note that you cannot simply place a doorbell switch or any dry contact in parallel and expect this to work.  

The long answer is that the 12Vdc coming from the garage opener itself is not just a DC voltage; there is a variable frequency PWM signal riding on top of this which communicates to an embedded STM8S103F2 microcontroller in the 882LMW.

I did not take my scope to the PWM signal, but my multimeter picked up a frequency range of ~6-14kHz.  The MCU likely uses input capture mode on a timer, and then returns the message with another PWM signal superimposed on the 12Vdc, or something like that.

Here is a picture of the back of the PCB.



My requirement is to make use of a computer controlled relay (dry contact) to open my garage.  I simple soldered two leads across the micro pushbutton and wired them to the relay.  My wires from the garage are now screwed to the terminals labelled red and white.

This works in parallel to the other 882LMW that's located in the entryway of my house.

I am not tinkering with this any further as I won't be able to reverse engineer the embedded firmware in the MCU, and it works the way I left it.