I did end up ordering the Axxess ASWC-1, plus a little adapter to fit the steeing wheel port in the back of the parrot radio.
The install was easy (four wires - power, ground, 2 canbus wires in the radio adapter). Making the connection was a little more annoying, as the ASWC instructions were close, but not 100%.
At the end of it all, I followed the install instructions, which required updating the firmware on the ASWC, and following a sequence of button pushes to connect it to the car. three tries and no love.
The device asks for the latest firmware on the head unit, so in the process of menu-hunting for the firmware version, I found an entry "steering wheel controls".
After enabling the radio to accept the steering wheel controls, all is working fine.
It works for the radio, will be checking if it adjusts in-call volume shortly.