Should I connect the AR6203 COM radio to the GMA324 audio panel using symmetric or asymmetric wiring?
Since you're talking about a DC power system, chassis ground and power ground should be the same. So the power ground is commonly used for grounding shields. The question is: should you ground one end of the shields or both?  

The purpose of the shields on the audio connections is to prevent induced audio frequency noise. To do that, the shields must have at least one ground point. RF engineers I've worked with will explain that the best solution is to have the shields grounded at both ends if the units at both ends are properly grounded and bonded. That's a big if, especially in general aviation avionics where slide-in mounting trays are used.  

While the connections may be good when you install them, over time the connections can degrade and resistances increase. When that happens the "grounds" at the connected units are no longer at equal potential which can induce current flow through the shields. This 'ground loop' can induce a lot of noise into the lines.  

Most installers I've worked with recommend terminating the shields only at one end thus eliminating any possibility of a ground loop. The key to this method is making sure any 'gap' necessary in the shield at the open end connector is as small as possible while still ensuring it doesn't make a connection. The small gaps aren't a big concern here as you're dealing with audio frequencies.