2 ACFT DEPARTING PARALLEL RWYS WITH ONE MAINTAINING VISUAL SEPARATION ON THE OTHER TURNS IN FRONT OF THE OTHER WHEN CTLR APPROVES L TURN. SECOND ACFT GETS A TCASII RA AND DSNDS. NMAC RESULTED.

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RPTRS ACFT DEPARTED FLL RWY 9L ON TWR ASSIGNED 090 DEG HDG. AT 1500 FT MSL; PNF OBSERVED CONFLICT ACFT HDG 360 DEGS ON COLLISION COURSE AND ALT. PNF ADVISED PF OF CONFLICT. PF EXECUTED EVASIVE MANEUVER TO PASS BELOW CONFLICT ACFT. EVASIVE MANEUVER WAS COMPLETED AT 880 FT MSL. PF THEN RESUMED NORMAL CLBOUT PROCS. PNF THEN CONTACTED FLL TWR TO ADVISE. TWR ADVISED THAT CONFLICT ACFT HAD DEPARTED SIMULTANEOUSLY FROM RWY 9R WITH CLRNC TO MAINTAIN VISUAL SEPARATION WITH RPTRS ACFT BEFORE COMMENCING NBOUND TURN. FOLLOW UP: RPTR SPOKE WITH TWR MGR ON THE FOLLOWING DAY. TWR MGR EXAMINED TAPES AND CONCLUDED THAT THE CTLR SHOULD HAVE ADVISED CONFLICT ACFT TO PASS BEHIND THE RPTRS' ACFT. TWR MGR ASSUMED RESPONSIBILITY AND ADVISED THE RPTR THAT THE SIT HAD BEEN DEBRIEFED AND CORRECTED. ALTHOUGH THE TWR MGR GRACIOUSLY ASSUMED RESPONSIBILITY; THE RPTR BELIEVES THE PF; THE CONFLICT ACFT; EITHER DISREGARDED HIS CLRNC OR LOST SIGHT OF RPTRS' ACFT. SUPPLEMENTAL INFO FROM ACN 352369: WE WERE TAKING OFF OF RWY 9L AT FT LAUDERDALE. OUR TKOF INSTRUCTIONS WERE FLY HDG 090 DEGS CLRED FOR TKOF. THE TKOF ROTATION WAS NORMAL; ROTATED TO 18 DEGS AND BEGAN CLBOUT. PASSING 1000 FT WE LOWERED NOSE CALL FOR 'FLAPS 1 DEG LEVEL CHANGE; 210 KTS' AS WE WERE ACCELERATING THROUGH 180 KTS. THE CAPT AND I NOTICED A SMALL ACFT AT THE 12:30 O'CLOCK POS; ON A COLLISION COURSE; SO I (THE FO) PUSHED THE NOSE OVER TO ABOUT 1500 FPM DSCNT AND CLRED THE ACFT BY 500 FT AND RECOVERED ACFT BY 1000 FT AGL. THE TCASII WAS CALLING OUT 'DSND; DSND; DSND' AND INDICATED RED BAND 3000 FPM DSCNT; AND THEN SAID 'MONITOR SINK RATE.' BEGAN THE RECOVERY AT 1000 FT 210 KTS. SUPPLEMENTAL INFO FROM ACN 352371: THE LAST LEG OF THE DAY WAS A REPOSITION OF THE ACFT FROM FT LAUDERDALE INTL ARPT TO OUR HOME BASE AT FT LAUDERDALE EXECUTIVE ARPT WHICH IS 8 MI N. ONLY THE REQUIRED CREW OF 2 PLTS WAS ABOARD OUR FAIRCHILD METRO 3 ACFT ON THIS LEG. I WAS ACTING AS CAPT WITH A QUALIFIED CAPT AS A FO DUE TO A RECENT INJURY TO 1 OF OUR COPLTS. ON THIS LEG I WAS THE PF; WITH THE COPLT HANDLING COMS. TWR CLRED US FOR TKOF ON RWY 9R AS STATED 'NXXXXX CLRED FOR TKOF RWY 9R; MAINTAIN RWY HDG AT OR BELOW 1500 FT.' IT WAS XX00 PM ESTIMATED AND DUSK WAS QUICKLY TURNING INTO NIGHT. NO REF TO TFC WAS MADE AT THIS TIME. AFTER CLBING THROUGH 1000 FT FOR 1500 FT; COPLT REQUESTED 'NXXXXX LOOKING FOR A L TURN.' AT THIS TIME A B737 HAD JUST DEPARTED RWY 9L. I LOOK OUT THE WINDOW AND ACKNOWLEDGED TFC IN SIGHT. COPLT ACKNOWLEDGED TO TWR 'TFC IN SIGHT NXXXXX.' TWR IMMEDIATELY SAID 'ROGER TURN L NOW HEADING N.' COPLT ACKNOWLEDGED 'ROGER TURN L HEADING N NXXXXX.' A QUICK CHK OF OUR TCASII BY THE COPLT SHOWED NO TFC IN THE IMMEDIATE AREA AS I EXECUTED A CLBING L TURN TO THE N. AS WE WERE TURNING THROUGH A HDG OF 020 DEGS A TCASII TARGET APPEARED SHOWING OFF OUR L SIDE SOON FOLLOWED BY A 'TFC; TFC' CALL. THE B737 WAS APCHING FROM BEHIND THE L WING. AT THIS TIME I EXECUTED A CLBING R TURN TO 1800 FT; AS THE B737 APPEARED TO BE DSNDING AND EXECUTING A TURN TO THE R. B737 RPTED MAKING EVASIVE MANEUVER. THE NEAREST VERT SEPARATION SHOWED ON TCASII WAS 300 FT BELOW US AND THE B737 PASSED APPROX 2000 FT BEHIND OUR L WING NOTED BY THE COPLT. THIS OCCURRED 1 1/2 MI OFF THE DEP END OFF RWY 9L. I BELIEVE 1 GRAVE ERROR WAS MADE BY 3 DIFFERENT PEOPLE. 'SITUATIONAL AWARENESS.' FIRST I THINK THE CTLR ANTICIPATED THE B737 TO ACCELERATE AND CLB MUCH FASTER THAN US THEREFORE IT SHOULD HAVE BEEN ABOVE AND PAST OUR POS WHEN HE SAID 'TURN L TO N NOW.' SINCE WE WERE EMPTY THE PERFORMANCE MAY HAVE SURPRISED THE CTLR. SECOND; I ALSO MISCALCULATED THE RELATIVE POS; IE; POS; VELOCITY AND CLB RATE SO I BELIEVED WE WOULD PASS BEHIND THE B737 INSTEAD OF XING IN FRONT OF ITS DEP PATH. THIS MISCALCULATION WAS CONFIRMED BY THE CTLRS STATEMENT 'TURN L HEADING N NOW.' THIRD; MINE AND MY COPLT'S LACK OF ORIENTATION OF PROGRESS OF TFC. THE B737 WAS XING OFF OUR L SIDE; WHICH MADE IT DIFFICULT FOR THE COPLT TO SEE.