B757-200 CREW BEGAN A TURNOFF OF RWY 13L INTO TXWY ZA. TXWY ZA WAS NOTAMED CLOSED.

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LAST FLT RETURNING TO STATION. NOTAM: TXWY ZA TURNOFF CLOSED. ISSUED IN JFK ARR ATIS INFO PRIOR TO OUR ARR. STATION ARR SPACING TO RWY 13L AT 5 NM. TFC AHEAD AND BEHIND US. WE EXPECTED THE NEED TO BE EXPEDITIOUS IN CLRING THE RWY. I FAILED TO REMIND CAPT OF THE CLOSURE OF TXWY ZA AFTER TWR CLRED US TO LAND. I NORMALLY TRY TO MAKE A POINT OF REMINDING THE PF OF ANY PERTINENT OPERATIONAL ITEMS FOR LNDG WHEN TWR ISSUES LNDG CLRNC. ON ROLLOUT; WE INITIATED THE TURN ONTO TXWY ZA AND STOPPED AS TWR CALLED TO INFORM US THAT THE TXWY WAS CLOSED. WE INFORMED TWR UPON THEIR QUERY THAT WE REQUIRED MARSHALLER ASSISTANCE OR TUG. THEY CLOSED RWY 13R AND SENT GND PERSONNEL TO ASSIST. THEY REMOVED THE TFC CONES AFTER INSPECTING THE TXWY AND MARSHALLED US ON TXWY ZA TO TXWY B. WE TAXIED TO OUR GATE. WHILE WAITING FOR ASSISTANCE; WE QUERIED GND WHY THE TXWY LIGHTS FOR TXWY ZA WERE ILLUMINATED. THEY INDICATED THAT THEY ARE HARD-WIRED TO THE SAME CIRCUIT THAT SERVES RWY 13L. WITH LIMITED VISIBILITY AT NIGHT; AND USING TFC CONES WITH REFLECTIVE TAPE AS A VISIBLE REMINDER/BARRIER; OPPORTUNITY OVERCOMES ONE'S OVERSIGHT WHEN EXITING THE RWY. AN ILLUMINATED SET OF BARRIER LIGHTS DURING NIGHT OPS WOULD OFFSET THOSE APPEALING ILLUMINATED LIGHTS OF A CLOSED HIGH SPD TXWY. SUPPLEMENTAL INFO FROM ACN 629296: THE HIGH SPD TXWY CTRLINE LIGHTS AND BLUE TXWY EDGE LIGHTS WERE ILLUMINATED SO I ELECTED TO TAKE THE HIGH SPD TXWY LIKE I NORMALLY DO ON RWY 13L. JUST AS I WAS ABOUT TO EXIT THE RWY; I WAS INFORMED BY TWR THAT TXWY ZA WAS CLOSED. SURE ENOUGH; THERE WERE UNLIT CONES BARRICADING THE TXWY. I HAD TO COME TO A COMPLETE STOP. I COULDN'T CLR THE RWY OR TURN AROUND. I HAD TO WAIT FOR WING WALKERS TO ASSIST ME. THE FLT BEHIND ME HAD TO GO AROUND; AND THE RWY WAS SUBSEQUENTLY CLOSED FOR 20 MINS UNTIL I COULD TAXI CLR. THE TXWY CLOSURE WAS NOT MENTIONED IN THE FLT PLAN NOTAMS BUT WAS REFED IN THE ATIS RPT. WE WERE FAIRLY BUSY DURING THE APCH AND MISSED THE SIGNIFICANCE OF TXWY ZA BEING CLOSED. WE SHOULD HAVE PICKED UP THE CLOSURE IN OUR APCH BRIEFING; BUT A QUICK ADVISEMENT CALL FROM THE TWR NOTIFYING US THAT THE HIGH SPD WAS LIT BUT WAS CLOSED AND UNUSABLE WOULD HAVE SURE HELPED.