CARGO MDT MAKES UNAUTH TKOF ON UNAUTH RWY.

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WE WERE OPERATING ON A SCHEDULED CARGO RUN. I WAS ACTING AS PNF (SIC) AND WE WERE CLRED TO CROSS RWY 23L AND TKOF ON RWY 23R. I ACKNOWLEDGED THE CLRNC AND THE CAPT CALLED FOR FINAL ITEMS. I COMPLETED THE FINAL CHKLIST WHILE WE WERE ROLLING DOWN THE RWY; MADE THE FIRST 2 CALLS AND LOOKED UP AND IMMEDIATELY NOTICED WE WERE ON RWY 23L! WAIT; WAIT; WAIT ISN'T PART OF THE STANDARD CALLS BUT THEN I CALLED; V1-R-V2 AND WE LIFTED OFF; FROM THE WRONG RWY. IT DIDN'T GO WITHOUT NOTICE FROM CLEVELAND TWR; WHICH INDICATED THAT IT WASN'T A PROB THIS TIME. WE WERE VFR AND THERE WAS NO CONFLICT WITH TFC. I QUERIED THE CAPT ABOUT THE INCIDENT AND WAS TOLD HE HEARD WE WERE CLRED TO TKOF; MADE THE TURN AND PUSHED UP THE THROTTLES. I BELIEVE HE REALIZED THE SIT AT V1 AT THE TIME AND CHOSE TO CORRECT COURSE OF ACTION BY TAKING OFF. BEING THE LAST LEG OF THE NIGHT; FATIGUE; LACK OF ATTN AND CLR UNDERSTANDING OF OUR CLRNC LED THE CAPT INTO MAKING A MAD DASH DOWN THE RWY WITHOUT QUESTIONING HIS SIC AS TO WHETHER WE WERE CLRED TO TKOF ON RWY 23L OR RWY 23R.