CONFLICT OCCURS BTWN A B737 DEPARTING AND AN ACR ON A MISSED APCH ON THE PARALLEL RWY AT LAS.

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JUST PRIOR TO OUR FLT; BEING CLRED FOR TKOF ON RWY 25R AT LAS; ANOTHER ACFT WAS SENT AROUND ON HIS APCH TO RWY 25L. WE WERE CLRED FOR TKOF AS THE PRECEDING B737 ROTATED AND WERE INSTRUCTED TO MAINTAIN VISUAL SEPARATION FROM ACR Y. THE CAPT WAS FLYING --INITIAL CLRNC WAS TO 7000 FT ON THE IDALE 2 SID. DURING THE CLB; AFTER TKOF DUTIES AND CHKLISTS WERE BEING ACCOMPLISHED. AT 5000 FT IN THE CLB; WITH ONLY THE PRECEDING B737 ACFT VISIBLE (THE MISSED APCH ACFT WAS BEHIND US NOT IN SIGHT); THE CTLR ISSUED A VECTOR; 'TURN TO 165 DEGS; CLB TO 6000 FT.' THE ACFT WAS LEVELING OFF AT 6000 FT IN THE L-HAND TURN WHEN THE CTLR REISSUED HIS VECTOR TURN TO HEADING 210 DEGS; WHICH NOW REQUIRED A ROLL TO THE R. AT THIS POINT; THE TCASII ALERT WENT OFF; 'CLB;' TFC DISPLAYS SHOWED THE ALERT ACFT WAS BEHIND US; WHILE THE PROCEEDING B737 WAS STILL IN SIGHT AND DEPICTED ON TCASII AS WELL. THE CAPT BEGAN A CLB TO SILENCE THE WARNING WHICH DID NOT HAPPEN UNTIL OUR ACFT HAD CLBED 500 FT IN ALT TO 6500 FT MSL. WE RPTED THE TCASII RESOLUTION TO ATC WHO THEN GAVE US A CLB TO 11000 FT AND A VECTOR TO RESUME THE IDALE 2 SID. AT THIS POINT; THE ATC CTLR INSTRUCTED US TO COPY HIS PHONE NUMBER AND HAVE THE CAPT CONTACT HIM AT THE COMPLETION OF OUR FLT. THE REMAINDER OF THE FLT WAS UNEVENTFUL. MY UNDERSTANDING IS THAT THE CTLR HAD THE CAPT CONTACT HIM TO APOLOGIZE FOR THE CTLR'S POOR VECTORS WHICH CAUSED THE MISSED APCH ACFT BEHIND US TO GIVE OUR ACFT A TCASII RESOLUTION.