AN ACR FLC ACCEPTED ANOTHER ACR'S CLRNC AS THEIR OWN AND TAXIED ONTO AN ACTIVE RWY CAUSING A COMMUTER ACFT TO GAR.

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FLT BTWN JFK DIRECT SFO ACFT BLOCKED OUT AT XA44 AT JFK. RWYS 13L AND 13R IN USE. CLRED TO TAXI L ON TXWY B; HOLD SHORT TXWY N FOR RWY 13R. APCHING TXWY U; GND CALLED AND WE ACCEPTED RWY 13L HOLD SHORT ON TXWY U. TWR CLRED A COMMUTER ACFT FOR TKOF RWY 13L AND ANOTHER ACR POS AND HOLD RWY 13R. WE TOOK THE OTHER ACR'S CLRNC AS POS AND HOLD RWY 13L AND FO READ BACK POS AND HOLD RWY 13R. THE OTHER ACR DID NOT READ BACK THE CLRNC BUT TOOK POS ON RWY 13R. 2 MINS LATER COMMUTER LNDG RWY 13L RPTED GAR DUE TO ACFT ON RWY 13L. TWR THEN SAW US ON RWY 13L AND CLRED US TO TAXI CLR TO THE L AND HOLD SHORT OF RWY 13L ON TXWY C. TOOK OFF ON RWY 13L AT XB08; LANDED SFO XE00 LCL. UPON ARR SFO CALLED JFK TWR CHIEF. HE STATED WE HAD TAKEN ANOTHER ACR'S CLRNC AND READ BACK RWY 13R. HE ALSO STATED THERE HAD NOT BEEN A LOSS OF SEPARATION AND NO VIOLATION WOULD BE FILED. I DID NOT HEAR THE CLRNC (POS AND HOLD RWY 13L); BUT ACCEPTED THE FO'S READBACK. I ALSO DID NOT PICK UP THAT HE READ BACK RWY 13R. HAD I ASKED FOR A RE- CLRNC FROM THE TWR AS I SHOULD HAVE THIS WOULD NOT HAVE HAPPENED. TWR HEARD OUR RWY 13R READBACK AND ASSUMED IT WAS THE OTHER ACR. THE OTHER ACR TAXIED ONTO RWY 13R WITHOUT A READBACK. TWR WAS WORKING BOTH RWYS 13L AND 13R ON ONE RADIO FREQ. TFC AT THAT TIME IS FAIRLY HVY. I COUNT AT LEAST 4 ERRORS THAT LED TO THIS INCIDENT.