SCT EXCHANGES TFC WITH ARRIVING B767. B767 ASSUMES TFC OBSERVED WAS PERTINENT TFC. B767 INITIATES TCASII EVASIVE MANEUVER FOR ACTUAL TFC DSNDING FROM ABOVE AND BEHIND.

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WE (ACFT X) WERE ON AN APCH TO RWY 25L AT LAX. APCH TOLD US THAT WE HAD TFC TO THE PARALLEL RWY AND THAT HE HAD US IN SIGHT. WE THOUGHT HE MEANT THE N COMPLEX AND IN FACT WE SAW TFC OFF TO OUR R FOR RWY 24R. AS WE DSNDED ON THE LOC/GS THROUGH ABOUT 2000 FT; WE RECEIVED A TCASII RA TO DSND AT 1500 FPM. AT THE LOWEST SCALE OUR TCASII SHOWED THAT WE HAD TFC DSNDING ON TOP OF US. AT 400 FT OR SO; I DECIDED TO TURN L TO GET OUT OF THE WAY AND WHEN I DID; THE RA ENDED. WE HAD TOLD THE TWR THAT WE WERE GOING AROUND AND ASKED IF WE COULD SALVAGE THE APCH. THEY SAID YES; AND I TURNED BACK TO THE RWY. IT WAS AT THAT TIME THAT WE SAW ACR Y TURBOPROP JUST OFF OUR R WING. WE HAD NO IDEA THAT THIS ACFT WAS EVER THERE. WE CONTACTED THE TWR BY PHONE ON THE GND AND THEY HAD US CALL TRACON. THE PERSON I TOLD THIS STORY TO TOLD ME THAT THEY WERE IN COMPLIANCE BY STATING THAT WE HAD PARALLEL TFC. I TOLD THEM THAT HAD THEY TOLD US THAT THE TFC WAS COMING UP FROM BEHIND US FOR RWY 25R; WE WOULD HAVE KNOWN WHAT WAS GOING ON.