Airbus 319 Captain reported a malfunction of the parking brake during taxi. The aircraft failed to stop when the parking brake was set. The Captain returned to the gate and the aircraft was taken out of service.

Narrative

Pushed out and got an ADR2 fault. Reset off gate with maintenance control. Continued taxi. Number X for Runway XXR; I set the parking brake and watched the triple indicator cycle. Looked at FO (First Officer) and felt the jet lurch forward and stop at slow speed. I check parking brake and ECAM. I then cycled the parking brake a few times and noticed that the time from my actuation of the switch to parking brake on ECAM was 8 seconds (about 4 times longer than usual). I repeated this several times and opted to return to a gate for maintenance to evaluate. I requested a tow in as I was unsure if the entire brake system was compromised and in their interest of safety went with the most conservative course of action. We blocked in and were met by three technicians and a supervisor. I was able duplicate the gripe for all of them. The jet was taken out of service.I have no theories as to why this happened. I do know that my poor night's sleep and fatigue level caused me to be slower to notice the trend in regards to a creep forward of the jet after the parking brake switch was engaged but prior to the actuators grabbing. I have filed a fatigue report as I called in fatigued after gate return.I do not have a solution or way to mitigate this type of systems failure.