Air Taxi First Officer reported; after engine start on the ramp; the aircraft began to roll forward resulting in the First Officer setting the parking brake.

Narrative

At approximately XA:20; after engine start the aircraft began to roll forward because the parking brake was not set. GPU was disengaged; no damage occurred; no ground personnel were injured and operations resumed normally following the embarrassing mistake. This was however; an event that could have; of course; resulted in far worse results; and one that I feared could be a possibility when asked to disengage the parking brake when leaving the aircraft in the ramp at the FBO. To elaborate on how the situation came into being; the crew was tasked to reposition the aircraft from an FBO to another FBO at ZZZ. After shutdown I informed the ground crew that we would be waiting for a few hours for our passengers to arrive. The ground crew asked us to disengage the parking brake in the event that the aircraft needed to be moved during that period. We were fully chocked and so I disengaged the brake; locked the plane; and went inside the FBO for the 3 hour wait. When preparing for passengers I didn't execute the flows I would've done for the first flight of the day and missed setting the brake then. The second attempt to catch this error was during the Before Start Checklist. This was one of those moments of looking; but not seeing". I looked at the Parking Brake annunciator; said "Engaged"; and carried on with the checklist. Not helping our situation was that the ground crew removed the chocks as we were going through the Before Start checklist. He approached and looked up at the cockpit as I signaled to keep us chocked; but either he misconstrued my signal or didn't really see what I signaled. Ultimately the chocks were removed and I didn't signal to return them.Following engine start; I signaled to remove the GPU and began going through flows with my head down. I then noticed some movement in my peripheral. What I saw was the lineman waving his hands alerting of our movement and I immediately applied brake pressure and engaged the parking brake then. I don't have a good reference for how far we moved; but I signaled to the lineman who was obviously flustered that our brakes were set with two closing fists. I then paused to make sure he and others were okay with a thumbs-up and he returned the sign. I called the FBO after the flight to apologize to the lineman; but he had already left for the day.Suggestions: This event highlights the importance of restarting a checklist when any interruption occurs. If I had done that I would have had an additional chance to catch the mistake before it could have resulted in error."