A320 Captain reported while attempting to park at the gate they didn't set the parking brake to allow brake cooling resulting in the aircraft rolling backwards. The Captain was able to apply the brakes stopping the backwards roll.

Narrative

Flight was uneventful from preflight thru landing. Due to a short landing roll and traffic on adjacent taxiways; I slowed the aircraft to make a Taxiway 1 turnoff. It became evident that we could make a XXL turnoff as well; so I slowed to taxi speed for that and as I started to ask the First Officer (FO) to get clearance for XXL turnoff off (ATIS only gave permission to use XXR) Tower gave us the clearance without asking as I think he saw what we were thinking and knew it would be better for ground traffic as well. It is important to note; I had never made that turnoff before; but due to the frigid temp; headwind; and light landing weight it was a smooth and controlled braking effort down to taxi speed. Taxi to gate XX was normal and with appropriate timing we shutdown Eng #2 and performed a Single Engine Taxi Without APU (SETWA) as all taxiways were clear. We stopped at the gate per the safe dock system. We hooked up power and shut down the #1 Eng and performed our parking flows. Once I saw 'chock on' I checked the brakes and saw them increasing in temperature; I then took the parking brake off and looked outside for a few seconds to confirm we weren't moving. When the FO finished her items; she did ask me if I wanted to set the parking brake and I told her 'the chocks are in and I will set it and take care of the fans when the temps stop climbing'. Apparently; around this point; either slightly before or after; we started moving backwards slowly; but I had no relative motion we were moving because airport operations (AO) was moving the jet bridge towards us at the time. We then felt a thump and I assumed it was the jetbridge hitting us. From my window; the jetbridge was at an angle and while the near corner was 5 ft away the other corner was angled out of my view and towards the aircraft so I thought it might have hit us. After a quick discussion with the FO; we knew something had happened. I set the Parking Brake and notified operations that AO may have hit the jetbridge or something to that effect. The AO crewmember looked frazzled; so I called operations for Maintenance and I think I told them to contact the safety department as well. Ground operations (GO) came on the headset and told us the aircraft had rolled back 'like 1 to 2 feet' I said 'weren't the chocks in' and he said they were pushed out of the way by the nose gear and that we also rolled over a chock or something to that effect. He confirmed the jet bridge did not hit us. After some time; Maintenance and GO looked over the aircraft; saw no damage; and then they hooked up a tug and towed us forward. The tow forward seemed more than just a few feet. The pax were kept well informed throughout the ordeal and they and the flight attendants were oblivious to any movement of the aircraft after our initial block in. Even despite the additional time of the Maintenance check and arrival the tow forward; the aircraft blocked in 6 minutes early.After block in; we were met by a ZZZ Assistant Chief Pilot (ACP) who saw the video and said---we rolled closer to 4 feet; that only the nose was chocked at the time; the chock was pushed away by the aircraft; and that the thump we felt was us rolling over one of the chocks the GO threw in front of it before the aircraft came to a stop. We also discussed my thoughts for releasing the parking brake for cooling. My rationale was simple; we did a quick stop in a short distance which would heat up our brakes for our quick turn. That being said; this ended up being poor situational awareness on my part. The same reasons we were able to stop so quickly on the runway would be the very same reasons our brakes wouldn't get very hot. Taking into account--light landing weight; frigid temperatures; a good wind; and an early arrival there was little chance our brakes would ever reach the 350deg threshold of the manual or affect our next departure. A post flight walk around revealed no damage to the aircraft. However; the following issues werenoted; a noticeable dip in the ramp and a 3 feet by 2 feet wet/deiced/sandy spot right where the nose gear should stop with safe dock. A log book entry was made; cleared; and we departed on time for our next leg.