Air carrier Captain reported after arriving at the gate and setting the parking brake; the A321 began to move forward. The Captain quickly held brakes and reset the parking brake. Captain does not remember if the brake triple indicator was pressurized or not. The aircraft was turned over to maintenance.

Narrative

Upon original gate had a check airman inform me this airplane had quirks as reported by previous Captain. She was deadheading to ZZZ. APU was on MEL. Upon gate arrival I set the park brake and started to release the brake pedals and quickly was distracted in my side window by the appearance of the jet bridge which looked like it was moving but something wasn't right. With the park handle set to on the jump seater noticed the brakes not engaged even though I had partial pressure applied with two engines running. The aircraft was moving not the jet bridge. Accumulator was fully pressurized. But no brake pressure was registered. The ramp agent was in a panic and I quickly did a scan and placed full pressure on brakes. I urgently signaled for chocks to be placed on nose gear. Both engine hydraulics should have been working (no yellow electric pump on). I recycled park brake and confirmed activation of brake pressure on triple indicator. Even though I remember seeing accumulator pressure; I don't recall seeing parking brake application applied on triple indicator before being distracted by movement and trying to process everything. I have flown the Airbus for years and have had a lot of things happen to me and quirks; but nothing like this. Kudos to the ground crew and Jumpseater for being amazing external resources. Aircraft was written up. No faults on ECAM or any other indications of something was wrong. Distraction during abnormal situation and ultimately even though we've been through a long day and trip with delays etc; make dang sure before releasing any pressure that park brake pressure is normal on triple indicator. Heavy reliance on things working as normal and expectation bias and complacency got me here. Even a seasoned Captain and Airbus pilot. I was taught to always reference the triple indicator when park brake is being applied and released. And I feel like over my years of flying the Airbus I've been pretty disciplined and constantly pointing as the dang thing. But it only takes once. One distraction. One time getting caught with not clearly processing events. I've had a few brake issues over the years. I simply knew better with Apu INOP and Both engines running to be on my game. But when something isn't right and you're hyper focusing on everything wrong it's hard to get back into the mindset. Thankfully the jumpseater (A320 CA (Captain)) helped by doubling the scanning and saw the issue. My only suggestion is to make awareness of brake issues and refocus on systems training and reinforce why it's so important to verify in totality the triple indicator…during Park Brake setting; Releasing; Gear Extension; etc. Maybe more pilots can learn from my event. Aircraft was written up and released to maintenance.