Air carrier pilot reported the parking brake failed while parking at the gate due to shutting down the incorrect engine.

Narrative

Arrived at the tow-in spot for gate XX; I set the parking brake and verified that the triple indicator was all in the green. I released the foot pedal brakes and shut down the motor. A moment later; I caught movement in my peripheral vision the aircraft was moving forward. I near simultaneously scanned the triple indicator; which showed zero; checked the parking brake; which was set; pressed the foot pedal again and recycled the anti-skid switch. The aircraft stopped. It all happened very quickly; and the aircraft moved forward. I called ground ops to check if they were okay; and they confirmed they were. After reaching the gate; we requested maintenance. When they arrived; I explained what had occurred and mentioned that I hadn't logged anything in the book because I didn't understand how it had happened or how to explain it concisely. They assured me they would handle it. Upon reflection later; I realized that I had shut down motor 1 first instead of motor 2. After further reflection later the next day; it seems right after the electrical transfer to the APU generator; yellow pressure was lost with the parking brake in the set position; resulting in the triple indicator brake pressure reading zero. Fatigue contributed to shutting down number 1 first and being out of sequence with a tow into the gate versus arriving at the gate and shutting down. I believed that maintenance would take the information I provided and make the necessary corrective actions and log book entry. I should have made the log book entry myself instead of delegating it to maintenance; however I was still in shock and stunned by what had just occurred and was unable to process what occurred until at a later time.