B737 FLT CREW IS ISSUED CLRNC TO MAINTAIN VISUAL ON TFC CLBING ON PARALLEL COURSE. TFC IS ACTUALLY CONVERGING AND B737 FLT CREW ALLOWS CONFLICT TO DEVELOP CAUSING TCAS RA.

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WE WERE BEING STEP CLBED BY DEP DUE TO TFC ABOVE US ALSO DEPARTING TO THE S. WHILE WE WERE LEVEL AT 11000 FT MSL; THE CONFLICTING TFC WAS POINTED OUT TO US AT 16000 FT MSL; 2 O'CLOCK POS AND ABOUT 10 MI. WE SAW THE TFC AND RPTED IT IN SIGHT. DEP THEN INSTRUCTED US TO MAINTAIN VISUAL AND CLB TO FL230. DEP ALSO TOLD US THE TFC WOULD BE TURNING TO A PARALLEL COURSE. THE FO (PF) CONTINUED THE CLB. AS I WATCHED THE TFC I NOTICED IT WAS NOT MOVING IN THE WINDSCREEN (IE; WOULD BE A CONFLICT IF SOMETHING DIDN'T CHANGE). DEP THEN INFORMED US THE TFC WOULD CROSS R TO L AHEAD OF US. AS THE TFC WAS STILL NOT MOVING IN THE WINDSCREEN; I TOLD THE PF TO SLOW THE CLB. THE PF DID NOT SLOW THE CLB AS HE WAS PLANNING TO CLB ABOVE THE TFC. I REMINDED HIM THAT WE WERE TO MAINTAIN A VISUAL AND THAT WE WOULD LOSE THE OTHER ACFT UNDER THE NOSE IF WE CONTINUED THE CLB. THE PF THEN REVERSED HIS CLB AND WE BEGAN TO TURN BEHIND THE TFC WITH ATC'S CONCURRENCE. AS THE PF ARRESTED HIS CLB AND BEGAN TO DSND AND TURN BEHIND AND BELOW THE TFC; WE GOT A CLB RA. OUR ACFT VECTOR WAS ALREADY BEHIND AND BELOW THE TFC AND TO COMPLY WITH THE TA/RA CLB COMMAND WOULD HAVE BEEN THE WRONG ACTION. AS THE TCAS CAUGHT UP WITH OUR ACFT VECTOR; IT DIRECTED A DSCNT WHICH WE WERE ALREADY IN. WE PASSED BEHIND AND BELOW THE ACFT WITH 1000 FT ALT SEPARATION. THE REMAINDER OF THE FLT WAS UNEVENTFUL. I COULD HAVE BEEN MORE ASSERTIVE IN MY DIRECTIVE COMMENTS TO THE PF. DEP SHOULD HAVE ENSURED THAT OUR COURSE WOULD IN FACT PARALLEL THE OTHER ACFT'S COURSE. SUPPLEMENTAL INFO FROM ACN 764272: WHILE CLBING OUT ON THE DEP WE WERE GIVEN TFC AT OUR 2 O'CLOCK POS AND WERE TOLD THAT HE WAS PARALLELING OUR COURSE. WE CALLED HIM IN SIGHT. WE WERE GIVEN INSTRUCTIONS TO MAINTAIN VISUAL CONTACT AND CLB FROM 11000 FT TO FL230. THE FO (ME) WAS FLYING AT THE TIME. WE WERE LIGHT (32 PAX) AND I BEGAN THE CLB. IN THE CLB; THE CAPT NOTICED THAT THE OTHER ACFT WAS STATIONARY IN THE WINDSCREEN (CONVERGING COURSE) AND DIRECTED THAT I LEVELOFF. AT THE SAME TIME; THE CTLR STATED THAT THE TFC WOULD NOW BE MOVING FROM R TO L. AT THAT POINT I BEGAN A DSCNT TO KEEP HIM IN SIGHT AND GET BELOW HIS ALT. AS I WAS DSNDING; WE GOT AN RA. I ALSO MANEUVERED BEHIND THE OTHER ACFT'S FLT PATH TO ENSURE FURTHER SEPARATION. WE PASSED 1000 FT BELOW AND WELL BEHIND THE OTHER ACFT. WE WERE CLRED BACK ON COURSE AND RESUMED OUR CLB. AT NO TIME DID THE CTLR ISSUE AVOIDANCE INSTRUCTION OTHER THAN TO ALLOW US TO MANEUVER BEHIND THE OTHER ACFT. IF WE WOULD HAVE KNOWN THAT OUR FLT PATHS WERE CONVERGING AND NOT PARALLEL; WE WOULD HAVE DELAYED THE CLB. THERE IS NO BLAME TO PLACE HERE. THE CTLR DID TELL US TO MAINTAIN VISUAL SEPARATION. I'M GLAD THAT HE SAW THE POTENTIAL CONFLICT IN TIME AND WERE ABLE TO RESOLVE IT WITHOUT INCIDENT.