Global 6000 First Officer reported they released the parking brake after parking to allow the brakes to cool down. After starting both engines the aircraft rolled about three feet.

Narrative

After landing at ZZZZ the right brake temp indicator was showing a 6. After confirming with ground crew that the aircraft was chocked it was decided we would release the parking brake to allow the brakes to cool. While running through the before start checklist; the 'parking brake set' line item was read and responded without actual verification. The parking brake was not set. After accomplishing both engine starts; both pilots were heads down in the cockpit accomplishing after start flows. I looked up and realized the aircraft was rolling uncomanded I quickly stepped on the brakes and altered the pic that we were rolling. The aircraft came to a stop and the parking brake was set. The aircraft had only rolled about 3 feet and was not in contact with anything around us. A Marshaller was standing in front of the aircraft but was still 10-15 feet away from the nose. As a crew we decided to run the before start checklist again as well as the after start checklist to ensure nothing else was missed. We talked about the reason for the Parking brake not being set and took a safety time out to ensure we were both okay to continue. Remainder of the flight continued without issue.