CE-560XLS flight crew crossed a runway without clearance and had a near miss with another aircraft.
Synopsis
CE-560XLS flight crew crossed a runway without clearance and had a near miss with another aircraft.
Narrative
We were not in anyway hurried or operating at an unreasonable pace. We landed on XXL ZZZ. We were unfamiliar on where the FBO was. There was a discussion to the tower operator and he says it's a bit confusing and he would give a progressive. As we cleared the runway XXL. and under the knowing we would be guided to the FBO; I let down my guard. I didn't hear the taxi clearance clearly; missing the hold short of XXR…a parallel runway quite close to the runway I was exiting from on taxiway 1. I cleared and tracked straight; looked left and I missed the hold short to make sure that area was clear and then before I could get the words 'clear left' clear to cross (same time bringing my head around to look at the opposite direction; I was stunned to see a small training aircraft (a diamond) climbing (estimated to be 200; feet or higher) overhead on the runway I thought we were cleared to cross without issue. My FO was 'heads down completing the after landing checklist and was surprised to see the same; not knowing I hadn't heard the clearances in its entirety and I was taxiing beyond the hold bars. I had chose to cross as stopping could potentially be more harmful if other planes were to follow. We realizes then we crossed an active runway without clearance. GND/TOWER was professional; and allowed us to progress to a safe location; communicated what we understood to have happened and gave us a telephone number; and then continued to give us the directions to the FBO.I called the telephone number given. Accepted responsibility without excuse and he gave me what his responsibilities were. Later; when we were ready to continue with our next trip; a further exchange of info occurred. (My name and contact info); and the rest of the trip and day went without further incidents.Suggestions: I admit that I fallen into a 'guard down state;' that finding the FBO wasn't going to be a search; and that in that I missed the part of the instructions (and read back) to hold short of XXL. Had I taken the time to communicate that to my FO; by stopping and asking. No matter how short the space between the runways; this would have avoided the incident. We also discussed and agree that no checklist is necessarily important enough it can't wait til cleared of all runways.
Second reporter narrative
After we landed on RWY XXL at ZZZ we were instructed to clear the RWY on taxiway 1 and Hold short of RWY XXR; we repeated the instructions back.After clearing the runway I performed the after landing flow and checklist. When I looked up from those duties; I saw we were already past the holdshort line for RWY XXR.Suggestions: Wait until clearing all parallel runways before performing any activities that can distract the Pilot monitoring.
Source: NASA Aviation Safety Reporting System (public domain). Reports are voluntary submissions and are not verified by NASA.