Seems like no one else had this issue, but I'll post this solution in case someone does in the future and comes looking for an answer:
What I was trying to do was midi learn the part volumes to separate knobs on the keyboard, so that I could be playing Part 1 and gradually bring up the volume of Part 2. I was able to midi learn my keyboard's knobs to the parts' volume knobs in Zyn's user interface, but turning them up or down had no effect on the sound.
At that time, I thought that midi parameter 7 (which is usually automatically assigned to volume) controlled the master volume for ALL parts. However, parameter 7 is actually assigned to part volumes... it just doesn't do anything to the knobs in the user interface. I had the Control 7 knob on the keyboard set as the global midi channel (00), so it was controlling all the parts and I thought it was Zyn's master volume.
So if you have, say
-part 1 assigned to midi channel 1
-part 2 assigned to midi channel 2
...you can assign one knob to Control 7, channel 1, and another to Control 7, channel 2 and POW, everything is great!
In short, the part volumes work fine, they are assigned to Control 7, and it's just the knobs in the GUI that don't do anything.