A320 Captain reported a parking brake failure when taxiing into the gate. The aircraft rolled forward under thrust and the left engine impacted the jet bridge.

Narrative

After landing at ZZZ; taxi into Gate XX (using SafeDock with GO at the switch) progressed normally to a stop. Using normal Captain's flow; I checked accumulator pressure; set the parking brake (I don't recall if I saw hydraulic pressure on the triple indicator); turned OFF the seat belt sign and switched Engine Master 2 to OFF. The First Officer noted 'I think we're moving; we're moving!'. At this time I noticed movement as well and applied immediate braking to the rudder pedals and stopped the aircraft. We maintained pressure on the rudder pedals to keep the aircraft from moving. I verified that the parking brake handle was ON; but there was no accumulator pressure or Parking Brake memo. I cycled the parking brake handle three times with no change. I coordinated with Maintenance Control and Ground Operations. This is when I learned that the #1 engine pylon had struck the bottom of the jet bridge. GO (Ground Operations) chocked the nose wheels. I started the APU; turned the Yellow Hydraulic Pump ON (to assure hydraulic pressure) and then shut down Engine #1. After engine shut down; we noted the Parking Brake memo was now present and with with pressure showing on the accumulator; we turned off the Yellow Electric Pump. This made me doubt my belief that I saw Yellow Accumulator pressure when I did my flow before setting the Parking Brake. We went outside to see any aircraft damage and noted a dent on the top of the #1 Engine pylon where it contacted the jet bridge. I made two write-ups in the logbook for parking brake failure and damage to the pylon.