Air carrier pilot reported the aircraft rolled forward at the gate resulting in the aircraft striking the jet bridge.

Narrative

Upon arrival at ZZZ; station operations informed us that due to lightning in the vicinity the ramp was closed and a PAATS (Park At Any Time System) truck would guide our parking. After parking; the PAATS driver asked us to verify brakes were not hot (they were not; but brake temperatures were still increasing post-landing) and that the parking brake was set (it was). After completing the parking checklist; thinking I had seen the chocks installed signal from the PAATS truck driver; I released the parking brake to slow the brake heating. Meanwhile; the jet bridge had been extended to the airplane; and door 1L had been disarmed and opened to deplane passengers. Shortly thereafter an alarm sounded in the jet bridge and I sensed movement outside my left side windows. I went back to door 1L to determine what was happening as the aft portion of the jet bridge was making contact with the aft portion of the open door. Based on the alarm in the jet bridge; I thought the movement I had sensed earlier was uncommanded movement of the jet bridge. But after talking to station operations personnel; we determined it was the aircraft that was moving; and that wheel chocks had not been installed.