NMAC BTWN 2 MLGS TURNING FINAL FOR PARALLEL RWYS.

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WHEN ON DOWNWIND ATC ADVISED US TO TURN R TO 210 DEG. AS SOON AS WE ROLLED OUT ON THE 210 DEG HDG; ATC TOLD US TO TURN L TO 090 DEG; DSND TO 3000 FT; AND ADVISE FIELD IN SIGHT. WHILE STILL IN THE TURN WE WERE TOLD TO CONTINUE L TO HDG 030 DEG AND ADVISE FIELD IN SIGHT. WHILE PASSING THROUGH 4500 FT AND 090 DEG I SAW THE FIELD AND ASKED MY FO IF HE ALSO HAD IT IN SIGHT. HE SAID HE COULDN'T SEE IT; SO I TOLD HIM TO KEEP THE TURN COMING BECAUSE IT LOOKED TO ME LIKE WE WERE GOING TO OVERSHOOT THE FINAL APCH. I ALSO ADVISED ATC THAT WE HAD THE FIELD IN SIGHT AND THEY CLRED US FOR A VISUAL APCH TO RWY 36L. WE DID OVERSHOOT FINAL; ALMOST TO THE 36R FINAL APCH COURSE. WHILE STILL IN THE TURN; ATC GAVE US A VERY AMBIGUOUS TA. HE SAID 'THAT TFC JUST OFF YOUR R SIDE IS FOR THE PARALLEL.' HE MAY HAVE GIVEN A LOCK POS; BUT I DON'T THINK SO. I LOOKED OUT AND SAW A TWIN TURBO-PROP 2-3 MI AHEAD OF US; LINED UP FOR 36R. I THOUGHT THIS WAS THE TFC HE WAS INDICATING AND I TOLD HIM 'TFC IN SIGHT.' APPROX 10 SECONDS LATER THE AIR TFC CTLR SAID 'THAT WAS REALLY CLOSE;' AND AT THE SAME TIME MY FO SAID THAT AN ACFT HAD PASSED OVER US BY ABOUT 200 FT. I NEVER SAW THE ACFT; BUT IT WAS APPARENT TO ME THAT HIS WAS THE TFC THAT HAD BEEN CALLED OUT TO US JUST A FEW SECONDS EARLIER. AFTER LNDG I WENT INTO OUR OPS TO CALL THE TRACON TO FIND OUT WHAT HAPPENED. WHEN I GOT TO THE PHONE; THE CAPT OF THE OTHER CONFLICT ACFT WAS ALREADY TALKING TO THE SUPVR. THE SUPVR ADVISED THE OTHER CAPT THAT SINCE WE WERE BOTH ON VISUAL APCHS THERE WAS NO PROBLEM AND HE ANTICIPATED NO FOLLOW- UP ACTION. WE BOTH LEFT IT AT THAT. THE OTHER CAPT TOLD ME THAT HE WAS ON VECTORS TO A VISUAL APCH TO RWY 36R AND HAD BEEN CLRED FOR A VISUAL APCH APPROX 10 SECONDS PRIOR TO OUR CONFLICT. HE HAD ALSO BEEN PLACED IN A POS FROM WHERE HE COULD NOT TURN TO FINAL WITHOUT A LARGE OVERSHOOT. HE APPARENTLY SAW US BEFORE WE SAW HIM AND SLID OVER THE TOP OF US DURING HIS TURN. NO EVASIVE ACTION WAS REQUIRED BY EITHER ACFT; BUT 2 OF MY PAXS MENTIONED SEEING AN ACFT PASS VERY CLOSE OVERHEAD. THERE ARE A NUMBER OF FACTORS THAT AFFECTED THIS VERY SERIOUS CONFLICT. ATC FAILED TO POS EITHER ACFT PROPERLY TO EXECUTE A TURN TO FINAL WITHOUT OVERSHOOTING. THIS WAS COMPOUNDED BY THE CEILING (5000 FT BROKEN) AND BY CHARLOTTE APCH POLICY OF KEEPING ACFT AT 9000 FT UNTIL ON DOWNWIND. BOTH ACFT WERE TALKING TO DIFFERENT CTLRS. BECAUSE OF THIS; NEITHER ACFT HAD THE CHANCE TO CLR FOR THE OTHER ACFT BY USING THE RADIO. I CALLED THE WRONG TFC IN SIGHT. THIS IS ONE OF THE GREATEST DANGERS WHEN OPERATING IN THE TERMINAL AREA. OUR PROBLEM WAS COMPOUNDED BECAUSE WE WERE IN A TURN AND BECAUSE OF THE AMBIGUOUS NATURE OF THE TFC CALL. NEITHER ACFT WAS TCASII EQUIPPED. WHILE OVER HALF OF OUR MLG FLEET HAS TCASII; NEITHER OF THESE ACFT WERE TCASII EQUIPPED. THE TCASII WOULD HAVE PROVIDED AN ALERT; AURALLY AND VISUALLY; WELL AHEAD OF ATC'S WARNING.