B737 CREW LNDG SEA IS ISSUED A VISUAL APCH CLRNC TO RWY 34L AND RPT UNABLE TO SEE ANOTHER B737 ALONG SIDE FOR RWY 34R.

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WE WERE CLRED FOR A VISUAL APCH FOR SEA RWY 34R AND A B737 WAS CLRED FOR A VISUAL FOR SEA RWY 34L. WE RPTED ACFT Y IN SIGHT AND KEPT THE ACFT IN SIGHT THROUGHOUT THE APCH (UNTIL THEY WENT MISSED). WE WERE NOT TOLD TO FOLLOW ANYONE. ACFT Y WAS FOLLOWING A B757. OUR VECTOR AND TURN ON TO FINAL PUT US RIGHT NEXT TO EACH OTHER. WE WERE NOT STAGGERED. ONE OF THE ACFT Y PLTS WAS OBVIOUSLY CONCERNED BY THE TONE IN HIS VOICE AND WE HEARD THEIR TCASII IN THE BACKGROUND. THIS REMINDED ME TO SWITCH OUR TCASII TO TA ONLY. AS I SWITCHED THE TCASII TO TA ONLY WE RECEIVED AN RA. WE COMPLIED AND WERE ABLE TO CONTINUE THE APCH TO A NORMAL LNDG. ATC ASKED US DURING THE APCH; 'DO YOU HAVE THE B757 IN SIGHT;' REFERRING TO THE ACFT Y WAS FOLLOWING. WE STATED THAT WE DID NOT. ACFT Y CALLED ATC AND SAID SOMETHING TO THE EFFECT 'HOW CAN ACR NOT SEE US WE'RE RIGHT NEXT TO HIM; WE'RE GOING AROUND.' THEY TURNED L AND DISCONTINUED THEIR APCH. I SHOULD HAVE SWITCHED THE TCASII TO TA ONLY EARLIER; BUT REALIZED MY MISTAKE AND THOUGHT WE RECEIVED A 'MONITOR VERT SPD' RA; WHICH WE COMPLIED TO; FELT CONTINUING THE APCH WAS APPROPRIATE.