It turned out the new part was bad!
I checked the fuses... fine.
I tested the circuit by using a jumper wire between the two larger connections in the wiring connector (that connect to the larger copper pins in the switch) and the brake lights came on, so that increased my confidence that the problem was with the switch. I returned the switch to O'Reilly for a refund and decided to go with the OEM part.
I also learned that the same switch is used in some Civics and Preludes, but when I called the Honda dealership, they couldn't look up parts for an Acura. I found the part (#36750-S04-003) at the Acura dealer for $75. With the part number, I called the Honda dealership and their price was MORE ($105)!!
It is kind of ridiculous how much they are charging for this part!
One helpful thing I learned along the way: there is a button, underneath a small cover, to the right of the shift knob, that releases the interlock and allows you to shift out of park if the computer does not sense that the brake lights are on. So, at least you can drive the car (without having to be towed), though you will likely be without brake lights.
So, I'm back to believing that the original problem was a brake light switch that was starting to go bad (likely some problem with the contacts within the switch). I hope it's solved, for good, now. I'm not interested in replacing that switch again! (even though it takes only about 10 mins to replace... even less now that I've done it a few times now!).