LCL CTLR TRANSPOSES ACFT NUMBERS CAUSING CONFUSION FOR ACR ACFT LNDG ON PARALLEL RWYS.

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OUR ACFT WAS FLYING THE SLC 159 DEG RADIAL TO INTERCEPT THE 34L LOC TO SLC APPROX 10 NM OUT. APCH CTL CALLED TFC AT OUR 10-11 O'CLOCK APPROX 6 MI. WE DID NOT SEE TFC ON HIS CALL AND RPTED SO TO APCH. AT APPROX 3-4 NM FROM RWY WE SIGHTED AN LGT Y TURNING L (AT OUR 1130 O'CLOCK) TO INTERCEPT FINAL TO 34L OR R. HIS R WING WAS UP SUCH THAT IT APPEARED HE DID NOT SEE US. WE EASED R (E) OF 34L/R CTRLINE EXTENDED TO INCREASE LATERAL SEPARATION. AS THE LGT Y MOVED TO OUR 9 O'CLOCK POS. APCH CTL WAS IN CONFUSING CONVERSATION WITH THIS OTHER ACFT AS WE APCHED APPROX 3 NM FROM RWY; SO WE SWITCHED TO TWR ON OUR OWN TO GET CLRNC TO LAND ON 34R. WE RECEIVED CLRNC TO LAND ON 34R AND DID SO SAFELY WITH NO CONFLICT. THE LGT LANDED ON 34L. THERE WAS NEVER ANY IMMINENT DANGER OF COLLISION; BUT HAD WE NOT SEEN TFC (HAZE; NIGHT; DUSK) THERE COULD HAVE BEEN. SUPPLEMENTAL INFO FROM ACN 216566: CLRED FOR VISUAL TO 34L. TCASII INDICATED CONFLICTING AND EVASIVE MANEUVER REQUIRED. APCH CTL CONFUSED AND COMPLICATED SITUATION BY GIVING INSTRUCTIONS TO TURN TO 280 DEGS WHEN ALREADY ESTABLISH ON BASE. IN FOLLOW-UP TELEPHONE CONVERSATION WITH APCH CTL SUPVR IT WAS LEARNED THAT CTLR USED WRONG CALL SIGN IN GIVING INSTRUCTIONS TO TURN TO 280 DEG. CONFLICT WAS RESOLVED BY INITIATIVES OF EACH ACFT TO AVOID EACH OTHER.