UNAUTHORIZED RWY ENTRY BY ACR.

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DFW TWR CLRED US INTO POS TO HOLD ON RWY 36R. SHORTLY THEREAFTER TWR CHANGED HIS INSTRUCTIONS AND TOLD US TO TAXI PAST 36R TO RWY 36L. BOTH MYSELF AND MY FO BELIEVED WE HAD BEEN CLRED INTO POS ON 36L. WE TAXIED ONTO 36L AND WERE TOLD WE DID NOT HAVE CLRNC ONTO THAT RWY. WE TAXIED IMMEDIATELY OFF 36L. SUPPLEMENTAL INFO FROM ACN 191697: WHILE I WAS MAKING MY ANNOUNCEMENT AND THE CAPT GETTING IN POS WE WERE ISSUED ANOTHER CLRNC TO CROSS RWY 36R TO 36L. AT THAT TIME WE WERE BOTH BUSY AND WE UNDERSTOOD THAT WE WERE CLRED IN POS AND HOLD ON 36L SO WE DID. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR PARTICIPATED IN STRUCTURED CALLBACK FOR RWY INCURSION STUDY. DOES NOT KNOW WHY DFW TWR MADE LAST MIN RWY ASSIGNMENT CHANGE; BUT AS THEY WERE GOING INTO POS ON 36R THEY WERE VERY BUSY WITH TKOF PREPARATIONS; AND IT IS LIKELY THAT THEY MAY HAVE MISSED ATC INSTRUCTION TO HOLD SHORT OF 36L. WHEN CTLR SAW THEM ENTER RWY 36L HE SAID THAT THEY SHOULD NOT BE ON THE RWY SO FLC IMMEDIATELY TAXIED BACK OFF RWY. DID NOT SEE OR HEAR ABOUT ANY CONFLICTING TFC. THEY LATER WERE INSTRUCTED TO USE 36R (THE ORIGINAL RWY) FOR TKOF. THINKS THAT WHEN LAST MIN CHANGES ARE MADE BY ATC THEY SHOULD EMPHASIZE THE CRITICAL INSTRUCTIONS.