MULTIPLE RWY OP PARALLEL RWYS ACR X NON ADHERENCE TO ATC CLRNC TCASII RA HAD LTSS FROM ACR Y. PLTDEV.

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ACR X TKOF LAX RWY 25R LOOP 9 DEP; WHICH CALLS FOR A 250 DEG HDG UNTIL XING SMO 160 DEG RADIAL THEN L TO 235 DEG HDG; CLRED ALT 2000 FT MSL. UPON SWITCHING TO DEP CTL VISUALLY ACQUIRED SMA FLYING S IN VFR CORRIDOR THEN GOT TCASII RA TO LOWER CLB RATE. FOLLOWED TCASII RA. RECLRED TO 4000 FT AFTER CLR OF SMA. IN CLB BTWN 2000 FT AND 3000 FT; REALIZED WE WERE STILL ON 250 DEG HDG; THEN TURNED TO 235 DEGS. TWR HAD CLRED ANOTHER ACR Y FOR TKOF FROM RWY 24L AFTER OUR TKOF (AND ON A SEPARATE TWR FREQ). ACR Y WAS BEHIND AND TO OUR R DURING INITIAL CLB AND NOT IN SIGHT. ATC GAVE US SUBSEQUENT CLBS AND VECTORS AS PART OF SID. WE WERE ASKED TO CALL DEP BY TELEPHONE UPON LNDG IN JFK. AFTER CALLING BY PHONE I WAS INFORMED THAT OUR FAILURE TO TURN UNTIL AFTER PASSING THE SMO 160 DEG RADIAL AND POST TCASII RA RESULTED IN A LOSS OF SEPARATION WITH ACR Y DEPARTING RWY 24L. I RECALL NO MENTION TO US OF ACR Y ON TWR OR DEP CTL. THE NEXT DAY MY FO TOLD ME HE HEARD US BEING CALLED AS TFC ONE TIME TO THAT RWY 24L ACR Y. CONCENTRATING ON THE SMA RA I FAILED TO HEAR OUR BEING CALLED TO ACR Y NOW ON DEP CTL. NATURALLY THE SID CALLED FOR A MORE PROMPT TURN TO 235 DEGS BUT CONCENTRATING ON AVOIDING THE SEEN SMA THAT PASSED 500 FT OVERHEAD DELAYED OUR RECOGNITION OF OUR FAILURE TO TURN PROMPTLY; PRODUCING A LOSS OF SEPARATION WITH ACR Y WE DIDN'T KNOW TO BE THERE; BEHIND US AND TO THE R ON A PARALLEL HDG. I WILL REDOUBLE EFFORTS TO AVOID FUTURE RECURRENCE. IT ALSO WOULD HAVE HELPED IF ATC HAD ADVISED US OF THE TFC AND REMINDED US OF NEED TO TURN. THE RWY 24L ACR Y WAS EVENTUALLY SEEN BY FO; BUT WE HAD ALREADY INITIATED TURN AT THAT TIME.