Air taxi First Officer reported after engine start on the ramp the aircraft rolled uncontrolled due to the parking brake not being set.

Narrative

The following is an event that took place on the FBO ramp at the ZZZ airport; in a Citation Sovereign 680 jet. Looking back on it; it reads like an accident chain that; by the grace of God; didn't end in an accident; and no one was injured. The Sovereign is a jet that is required to operate with two pilots. Both pilots on this flight were type-rated and qualified to act as PIC. Our company had designated the other pilot as PIC for this two day trip. He flew the three legs on the first day. I was going to fly the two legs scheduled on this day - the second day - flying from the left seat. This arrangement was not at all unusual; and had nothing to do with what happened.Accident link #1 - We arrived at the FBO in plenty of time - about 1:15 before scheduled departure. We both worked on getting the aircraft ready. Preflight inspection; connecting the nose gear steering link; getting ice for the cooler and hot coffee for our passengers. I asked the PIC as to when he wanted me to start the APU and begin the cockpit setup. He said to start it :20 minutes before our scheduled departure time. In my opinion that was not early enough; but I didn't object. I felt it was within his purview to make that decision. Nonetheless; I did start it a couple of minutes early. Accident link #2 - After I started the APU I began our cockpit setup. Checking all the various systems and configuring for our flight. At this point the APU had been operating for about :05 minutes. About then the PIC called me and said the passengers are here. Now I knew we were going to feel rushed. I interrupted my cockpit setup flow and tuned in the ATIS and wrote that down; in preparation for getting our ATC clearance to ZZZ1.Accident link #3 - Our flight plan had been filed using Foreflight. The PIC had filed it; choosing the route recommended by the Foreflight app. Their recommended route works perfectly about 99% of the time; with only occasional minor changes. Normally one would contact ZZZ Clearance Delivery; but in this case ATIS advised to get the IFR clearance on Ground Control frequency; XXX.X I requested our clearance on that frequency. The controller was very busy. He advised said I would be getting 'a full route clearance' and to advise when I was ready to copy. I already had a pad and a pen and so I told him to go ahead. He proceeded to rattle off a completely different route than what we had filed with perhaps five or six intersections; without spelling their names; as well as a new SID and an airway. He spoke so fast I had to scribble and abbreviate in an attempt to keep up. I read it back barely. About that time our passengers were led out to the plane by the PIC; and I felt like I was way behind. In addition; I quickly realized that there was no way I could put our ATC clearance in our FMS's or iPads without knowing the spelling of those intersections. So I had to call Ground Control back and ask for the proper spelling. And then I had to put the pretty complicated revised routing in both the FMS's and my iPad. The PIC had; by now; sat down in the right seat and he programmed his iPad and tried to help me get finished in our setup. By now the passengers had been onboard for :05 to :10 minutes and we both felt the pressure of time.Accident link #4. The PIC pulled out the Before Start checklist and we started to run it; planning to use the checklist to catch anything we might have missed. But we rushed the checklist too; with him asking the next 'challenge' before I had completely answered the previous item. We were both allowing the pressure of time to affect us; and it was about to have very serious results!As I started the first engine (the right side); I noted there was no marshaller present; but we had encountered that situation many times; and I wasn't concerned. After that engine was stabilized at idle; I conducted a flight control / spoiler check; and proceeded to start the left engine. During that start a marshaller came out. With the second engine stabilized; I called for the 'After Start' checklist and we went heads down as we performed our checks and verified our systems functionality. It was then I felt something in the seat of my pants; and looked outside to see that we were rolling ! And we had already rolled about twenty feet !! The Parking Brake had never been set !! The marshaller had moved off to the side. I regained control of the aircraft and steered it in a 90 degree turn to the right to exit the ramp. I stopped the airplane at the end of the ramp so we could pull ourselves together. We took a few minutes to get calmed down and make sure everything that needed to be done; was done. We were both shaken up; and we slowed way down and checked and double-checked everything. Once we leveled in cruise; we debriefed this whole event; and talked about how lucky we were that an accident had not occurred and vowed that neither of us would allow time pressure to cause us to rush things; not ever again ! Lesson learned !! We both knew better !