ACFT INBOUND TO MEM EXPERIENCED TCASII RA WITH PARALLEL RWY TFC.

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ARR VIA HOLLY-1 TO MEM. AFTER PASSING HOLLY SPRINGS WE WERE ASSIGNED 5000 FT AND A 280 DEG HDG 'VECTORS TO RWY 36R.' WE WERE GIVEN FURTHER DSCNT TO 3000 FT AND TOLD TO RPT 'ARPT IN SIGHT' FOR A VISUAL APCH. NOTE: WE DID NOT CALL ARPT IN SIGHT NOR WERE WE CLRED FOR ANY APCH AT THAT TIME. WE WERE GIVEN FURTHER DSCNT TO 2000 FT WITH NO HEADING CHANGE. DSNDING FROM 3000 FT TO 2000 FT (STILL ESTABLISHED ON THE 280 DEG ASSIGNED HEADING) APPROX 15 MI FROM THE ARPT; WE PICKED UP THE RWY. WE ATTEMPTED TO CALL 'ARPT IN SIGHT' BUT OUR XMISSION WAS BLOCKED. SUBSEQUENT ATTEMPTS TO GET A WORD IN 'EDGEWISE' WERE DIFFICULT DUE TO FREQ CONGESTION. AS WE WERE APCHING THE RWY 36R FINAL WE SAW ANOTHER ACFT (VISUAL AND TCASII) ON APCH TO RWY 36L. SHORTLY THEREAFTER WE GOT A TCASII CLB RA WHICH WE RESPONDED TO IMMEDIATELY. THE OTHER FLT THEN TOLD THE CTLR HE WAS RESPONDING TO TCASII RA (DSCNT). THE CTLR THEN SAID OUR FLT WAS JOINING THE RWY 36R LOC. AT THAT POINT WE DID MAKE THE TURN TO JOIN THE RWY 36R LOC AND THE RA WENT AWAY. THE CTLR ASKED IF WE HAD THE RWY. WE RESPONDED THAT WE HAD JUST COMPLIED WITH A TCASII RA (CLB) AND THAT WE DID HAVE THE RWY IN SIGHT. HE THEN CLRED US FOR A VISUAL APCH TO RWY 36R AND TO CONTACT THE TWR. THE REST OF THE APCH/LNDG WAS UNEVENTFUL. IN MY OPINION FACTORS THAT CAUSED THIS POTENTIAL CONFLICT WERE: FREQ CONGESTION -- 1 CTLR WORKING MULTIPLE FLTS TO MULTIPLE RWYS. CTLR PERHAPS THINKING HE HAD CLRED US FOR A VISUAL AND OR INTERCEPT; WHEN IN FACT THAT DID NOT HAPPEN UNTIL AFTER WE TURNED INBOUND AS WE HEARD HIM TELL THE OTHER FLT THAT WE WERE JOINING THE LOC.