CE-560 First Officer reported aircraft movement after parking brakes were set.

Narrative

After a normal engine start; with the parking brake still set; both crew members were heads down in the cockpit. We were getting clearance and setting up the cockpit. We both felt movement of the airplane at the same moment and looked up and the plane had moved; we quickly got on the pedal brakes to stop the aircraft. There was a lineman that was near the nose of the aircraft giving us a stop sign for a moving aircraft. The aircraft came close to them and could have caused an incident without our quick reaction. We had no intention of moving at that time; and still did not move for a few moments after due to setting up the cockpit and the stop sign from the lineman. The parking brake was trusted to do its job; and was checked multiple times prior to start; but at that moment; it did not. The lessons learned: keep your feet on the brakes even when the parking brake is set in case the brake gives way; and one pilot should be heads up at all times. I plan to implement both of these moving forward.