AN MD80 CREW MISINTERPED AFTER LNDG TAXI CLRNC AND TAXIED ACROSS THE PARALLEL WHEN NOT CLRED TO DO SO; AT SEA.

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I WAS THE CAPT OF FLT XXXX DFW-SEA. THE TIME OF THE INCIDENT WAS APPROX XB40; AND THE WX CONDITIONS WERE 1 OVCST WITH RVR'S OF 1000 FT TOUCHDOWN; 1000 FT MID; AND 1200 FT ROLLOUT. AFTER LNDG FROM A CAT III APCH; DISCONNECTING THE AUTOPLT; AND EXITING ON TXWY N; MY FO READ BACK TO TWR 'CLRED TO CROSS RWY 16L' AND I REPEATED ALOUD 'CLRED TO CROSS.' IN REALITY; ACCORDING TO THE FAA TWR; THE CTLR SAID 'HOLD SHORT OF RWY 16L' AND NEVER CORRECTED MY FO'S READBACK. THE MISTAKE WAS REALIZED AFTER WE HAD CLRED RWY 16L AND AN AIRPLANE TOOK OFF BEHIND US. I FEEL THE MISTAKE I MADE WAS NOT STOPPING THE ACFT BEFORE XING RWY 16L AND RECEIVING CONFIRMATION OF THE CTLR'S CLRNC BEFORE PROCEEDING; ESPECIALLY WITH A VERY DEMANDING APCH; EXTREMELY LOW VISIBILITY; COCKPIT DISTRS; IE; AUTOPLT WARNING OF DISCONNECTING AND THE EFFECTS OF ARRIVING AT XC40 BODY TIME. SUPPLEMENTAL INFO FROM ACN 498875: AS THE CAPT CALLED FOR FLAPS AND AFTER LNDG CHKLIST; TWR XMITTED INSTRUCTIONS FOR TAXI. I BELIEVED THE INSTRUCTIONS TO BE CLRED TO CROSS RWY 16L TO THE RAMP. I READ BACK THOSE INSTRUCTIONS AND WE PROCEEDED ACROSS RWY 16L.