I'm wondering if commercial passenger planes, like the A320, are always connected to the ground power unit (GPU) when they are at the gate? Even if for a quick turnaround.
Being connected to ground power is the usual case, but there are exceptions. Typically, the most common reason to not be connected is if the ground power unit at the gate isn't working. (Because aircraft use 400HZ AC current at some fairly high amperage, you can't just plug in directly to the standard electrical mains -- a dedicated Ground Power Unit is required, which incorporates some safety features so that equipment isn't damaged even if the unit is unexpectedly disconnected.)
Other reasons could be in the case of a quick turn, the pilots signal to the ground crew not to bother. Also, if high winds prevent the ground crew from connecting the large yellow air conditioning hoses to the aircraft, then the jet's APU will be running anyway to provide air conditioning, so there's no great need to connect & later disconnect the ground power... the pilots just let the APU power the electrical system as well at this point, saving the ground crew a couple of steps & a couple of minutes.
All of this is less common than the alternative case, though, where the ground power and ground air are connected and the APU is shut off. The air & electricity from the gate are cheaper than running the APU to provide those, so most often we prefer to turn off the APU until we get close to time to push back.