A320 Captain reported difficulties during gate arrival using a self-park" system. The aircraft stopped short of the gate and rolled several feet before the tug driver secured the aircraft to be towed the remaining distance."

Narrative

Uncommanded Aircraft MovementArrived in ZZZ from ZZZ1 for fuel and catering. Followed the Self Park system in to gate XX; received the 'STOP' sign followed by the 'OK' sign. We accomplished the Shutdown Checklist per SOP and noticed the Jetway Driver not willing to move the jetway. This being; for a lack of a better term; an irregular operation I was thinking the Jetway Driver wasn't sure if he should proceed to put the jetway up to the aircraft to open the main cabin door. I gestured for him to do just that as we needed to be fueled and his hand signals back were confusing; but clearly there was some sort of conflict with the ramp.A few moments later the ground crew plugged in a headset to advise us that we were just a little short of the stop line so they needed to tow us the last little bit in order to line up with the jetway. I explained that we had passengers standing in the aisle; etc. and that I would get back to him when we were ready. The passengers were quickly back in their seats and I informed the ground crew that we were ready. The Ramper said they were ready and asked me to release the Brakes; I released the Brakes and verbalized that they were released. I felt a sensation of the aircraft rolling backward momentarily but couldn't tell if it was the jetway moving; or the the aircraft moving. The Ramper came back on the headset and quickly asked me to set the Brakes. Sensing urgency in his voice; I immediately set the brakes and asked what was going on. He explained that the aircraft rolled backwards because he didn't have it secured. No damage or injury was incurred during the Uncommanded Aircraft Movement. Upon debriefing with both the Ramper over the headset and the Jetway Driver in person I learned that they've been having issues with the Self Park system at gates XX and XY with aircraft ending up short of the stop line even though the system shows 'OK'. Combine this with the fact that station ops will not allow the Jetway Driver to move the jetway until the aircraft is exactly on the stop line creates a degradation in safety with needing to coordinate to move the aircraft; literally inches; in order to position the jetway. I feel that there is a safer way to do this by either re-calibrating the Self Park system; allowing a little variance in jetway movement; or at the very least revert to a Marshaller until such time as the Self Park system is corrected.