CAPT OF AN MLG FAILED TO HOLD SHORT OF A PARALLEL RWY DURING TAXI TO GATE.

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I WAS CAPT ON FLT; AUG/XX/97; DFW-ATL. AFTER LNDG AND CLR OF RWY 27L I UNDERSTOOD A CLRNC TO PROCEED TO THE W END OF RWY 27R AND CROSS. I PROCEEDED TO THE W END OF ATL RWY 27R; TURNED N LOOKING EASTERLY DOWN THE RWY TO INSURE AGAINST CONFLICT AND CROSSED THE DEP END. AFTER ARRIVING AT THE GATE I WAS DIRECTED TO CALL ATL FAA TWR. I WAS ASKED WHAT I UNDERSTOOD TO BE MY TAXI CLRNC. I INDICATED MY UNDERSTANDING THAT I HAD BEEN CLRED TO CROSS AT THE W END OF RWY 27R. HE INDICATED THAT THE TWR CTLR HAD ISSUED A 'HOLD SHORT CLRNC' BUT HAD NOT RECEIVED OR INSURED A 'HOLD SHORT' ACKNOWLEDGEMENT FROM MY ACFT. SIMPLY 'RWY 27R AT THE W END ROGER.' HOW AROSE: MISUNDERSTANDING BTWN TWR AND COCKPIT. CONTRIBUTING FACTORS: POST LNDG WORKLOAD AND DEP/ARR WORKLOAD. CORRECTIVE ACTION: PLTS: INSURE CLRNC BEFORE XING. CTLR: INSURE ACKNOWLEDGEMENT ON READBACK. HUMAN PERFORMANCE: DOWNLOAD DUTIES AT CRITICAL TIMES. HEADS OUT DURING CRITICAL ARPT AREAS. SUPPLEMENTAL INFO FROM ACN 378470: RECEIVED CLRNC TO GO TO THE W END OF RWY 27R. I RESPONDED TO THE CLRNC AND PROCEEDED TO DO MY AFTER LNDG DUTIES. I GLANCED UP AS I THOUGHT WE WERE APCHING RWY 27R; BUT WE WERE IN FACT XING THE RWY. I CLRED THE APCH END OF THE RWY; TO MY R; AND SAW NO ACFT ON THE RWY FOR TKOF. I LOOKED BACK AT THE CAPT WITH A QUESTIONING LOOK WHICH OBVIOUSLY RAISED DOUBT IN HIS MIND. HE ASKED; WE WERE CLRED TO CROSS AND I SAID; I DON'T THINK SO; BUT AT THAT POINT WE WERE ALREADY ON THE RWY. LESSONS LEARNED: INSURE THE OTHER CREW MEMBER AS WELL AS THE CTLR POSITIVELY CONFIRM HOLD SHORT OR CROSS RWY CLRNCS.