AN SF340 FLC DEPARTS FROM AN UNAUTH RWY AT HOU; TX.

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CONFUSION AT A MULTI-RWY ARPT LED TO TAKING OFF ON A SHORT RWY THAT WE HAD NO PERFORMANCE DATA FOR. THE TKOF WAS WITHOUT INCIDENT. CONFUSION RESULTED FROM SEVERAL FACTORS. LNDG THE NIGHT BEFORE; THE TWR TRIED TO SWITCH US FROM THE R PARALLEL TO THE L WHEN WE WERE ON SHORT FINAL. I REFUSED THE SWITCH BECAUSE I WAS UNFAMILIAR WITH THE SHORT RWY; AND COULD NOT PULL OUT CHARTS. AFTER A STAND UP OVERNIGHT (ON DUTY ALL NIGHT); WE DEPARTED THE NEXT MORNING IN THE DARK; AT AN UNFAMILIAR ARPT; AND 4 HRS OF SLEEP. THE ATIS RPTED DEPS ON A LONG; FAMILIAR RWY. ON TAXI; ANOTHER ACR WAS ASSIGNED A SECOND RWY FOR DEP. WE WERE THEN ASSIGNED A THIRD RWY; DIFFERENT FROM THE ATIS RPTED RWY AND THE RWY OF THE PREVIOUS ACFT. WE WERE ASSIGNED THE PARALLEL RWYS FROM THE NIGHT BEFORE. I TAXIED TO THE LONGER PARALLEL WE HAD LANDED ON; WHICH THE FO AND TWR CONFIRMED WAS THE WRONG PARALLEL. AT THIS POINT; I TAXIED TO THE ASSIGNED PARALLEL; WHICH I ASSUMED WAS THE LONGER ONE WE HAD LANDED ON THE NIGHT BEFORE. I ASKED THE FO TO CHK THE PERFORMANCE DATA; AND HE SAID TKOF WAS ALLOWED. AFTER TKOF; I REALIZED WE HAD TAKEN OFF ON THE SHORT RWY; WHICH WE HAD NO DATA FOR. THE FO AND I HAD BOTH BEEN CONFUSED -- ME WITH THE RWY; HIM WITH THE CHART. FACTORS THAT AFFECTED US: FATIGUE FROM THE STAND UP OVERNIGHT. THE TWR'S CONSTANT EFFORTS TO ASSIGN US THE SHORTEST RWY FOR TKOF AND LNDG WHEN NOT LISTED IN THE ATIS. NORMALLY; I WOULD HAVE STOPPED THE ACFT DURING TAXI AND SORTED THIS OUT. I BELIEVE FATIGUE LED ME TO NOT THINK THIS THROUGH CLRLY. AS PIC; IT IS MY SOLE RESPONSIBILITY TO SORT THIS OUT BEFORE TKOF.