B737 EXITED THE OUTER OF PARALLEL RWYS AND WAS INSTRUCTED TO HOLD SHORT OF THE PARALLEL RWY. SUBSEQUENT INSTRUCTIONS FROM TWR WERE MISINTERPRETED AND THE PF CROSSED THE 'HOLD SHORT' LINE BEFORE COMING TO A STOP.

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I WAS THE PF AS THE CAPT. UPON LNDG ON RWY 24R AT LAX WITH THE WX CLR ON ROLLOUT; THE TWR CTLR MADE 1 CALL FOR COMPANY NUMBER TO CLR AT TXWY AA AND HOLD SHORT OF THE RWY 24L. THE FO FAILED TO ACKNOWLEDGE THE RADIO CALL AND I DIRECTED HIM TO RESPOND TO THE CALL. BEFORE HE COULD RESPOND; TWR CALLED US AGAIN; AND THIS TIME I THOUGHT THE TWR TOLD US TO EXPEDITE OFF THE RWY TO CROSS RWY 24L ON TXWY AA. I WAS TURNING OFF OF RWY 24R ONTO TXWY AA AND THE FO WENT OVER TO OPS FREQ. I DIDN'T CONFIRM THIS WITH FO EVEN THOUGH; IN HINDSIGHT; HE REPEATED WHAT THE CTLR TOLD HIM. FOR SOME REASON; FOR THOSE FEW SECONDS I WAS UNDER THE IMPRESSION WE WERE TO EXPEDITE THE XING OF RWY 24L AT TXWY ZZ. I PROCEEDED TO MOVE ALONG TXWY AA TOWARD RWY 24L. FOR SOME REASON I THOUGHT THERE WAS CONFUSION ON THE CLRNC AS I APCHED RWY 24L BUT WAS COMFORTABLE THAT I WAS CLRED TO CROSS. I LOOKED BACK TOWARD RWY 24L TO CROSS AND SAW AN ACFT ON SHORT FINAL. I ASKED THE FO TO CONFIRM WE WERE CLRED TO CROSS AND HE SAID HE THOUGHT WE WERE TO HOLD SHORT AND CALLED TWR TO CONFIRM. TWR CONFIRMED WE WERE TO HOLD SHORT. BY THIS TIME I HAD ROLLED A FEW FT PAST THE HOLD SHORT LINE BUT 10-20 FT OF ENTERING THE RWY ENVIRONMENT. I TOLD THE FO TO CALL TWR AND TELL THEM WE HAD CROSSED THE HOLD SHORT LINE ON TXWY AA. TWR ACKNOWLEDGED THE CALL; DIRECTED THE ACFT ON SHORT FINAL TO GO AROUND; AND AFTER SEVERAL MINS CLRED US TO CROSS RWY 24L. WE WERE GIVEN THE PHONE NUMBER TO TWR AND ASKED TO CALL. I CALLED THE TWR SUPVR SHORTLY AFTER PULLING IN TO THE GATE. I SHOULD HAVE VERBALLY CONFIRMED THE DIRECTIONS FROM TWR IN A MORE CONSISTENT MANNER WITH THE FO. I SHOULD NOT HAVE ALLOWED THE FO TO BE OFF THE FREQ AT A CRITICAL TIME AND I SHOULD NOT HAVE TAXIED OVER THE HOLD SHORT LINE UNTIL I WAS 100% CERTAIN I WAS CLRED TO.