Air carrier pilot reported after push back the aircraft rolled during engine start even though the parking brake indicated it was on.

Narrative

Flight (ZZZ-ZZZ1) pushed back from the gate with clearance from Ramp. Communicated completion of push with standard phraseology. CA set the PARK BRK handle to the ON" position; scanned accumulator pressure in the green and PARK BRK indicated on ECAM before replying with standard phraseology to the TUG driver that the PARK BRK was "ON". The flight crew planned a SET and opted to wait for wave off to start Engine 1 due to the anticipated lengthy delay time to taxi out of the ramp area. During engine start; an inbound aircraft was entering the ramp at Taxiway 1. CA noticed the inbound aircraft and sensed own aircraft movement; initially thinking it was a parallax / relative motion illusion. PARK BRK was indicated on the ECAM but the Triple Gauge Accumulator needle was outside of the "green". CA initiated the toe brakes and stopped aircraft movement. While overall movement seemed slight from the flight crew's perspective; it was enough to alert the Ramp Controller to query; and the inbound aircraft to stop. CA re-cycled the parking brake and verified accumulator pressure "in the green" and ECAM indicating PARK BRK. PARK BRK seemed operational and a scan of the HYD system page indicated systems normal. Crew decided to continue; but had to return to gate due to critical MOT (Minimum Off Time). PARK BRK was added to the Aircraft Logbook for further investigation by Maintenance."