A DEPARTING ACR HAS NMAC WITH ANOTHER DEPARTING ACR OFF PARALLEL RWY.

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I WAS THE FO ON FLT ACR X SCHEDULED SVC FROM ST LOUIS TO OKLAHOMA CITY. AT PM33 CST WE TOOK OFF FROM LAMBERT FIELD ON RWY 30L WITH A CLRNC TO 5000 FT AND HDG OF 305 DEGS. THE CAPT WAS FLYING THE ACFT. AT 1000 FT AGL I SELECTED FLAPS 1 AND SET CLB THRUST AT THE CAPT'S DIRECTION. SHORTLY AFTER THIS (AT AN ESTIMATED ALT OF 1300-1500 FT AGL AND ON THE ASSIGNED 305 DEG HDG) WE GOT A TCASII WARNING; THE IVSI DISPLAY WAS RED AND THE TCASII VOICE BEGAN COMMANDING 'REDUCE CLB.' THE CAPT STARTED TO LEVEL THE ACFT AS WE BOTH LOOKED FOR THE CONFLICTING TFC. I LEANED CLOSER TO THE R WINDOW TO CLR THE OTHER DEP (RWY 30R) AND SAW AN MD80 ABOUT 100 FT BELOW AND CLBING INTO US FROM R TO L. HIS LEFT WING WAS ALREADY MOVING UNDER OUR ACFT. I SAID SOMETHING LIKE 'KEEP CLBING; I HAVE HIM IN SIGHT DO YOU WANT ME TO TAKE IT.' WE TRANSFERRED CTL AND I PULLED THE ACFT UP TO GET VERT SEPARATION AND THEN TURNED RIGHT AWAY FROM THE MD80'S FLT PATH (HE WAS NOW PASSING UNDER US). WE RPTED THE INCIDENT TO TWR AND ASKED FOR INSTRUCTIONS. WE WERE GIVEN A HDG OF 060 DEGS; THEN 090 DEGS AND TOLD TO LEVEL AT 3000 FT MSL. THE MD80 STARTED A TURN BACK INTO US AND THE CAPT TOOK THE ACFT AND CONTINUED A TURN TO THE R. WE WERE VECTORED BACK AROUND ON COURSE AT 3000 FT. THE REST OF THE FLT WAS UNEVENTFUL. WE RECEIVED NO EXPLANATION FOR THE CONFLICT FROM TWR ALTHOUGH I HAVE SINCE BEEN TOLD THE MD80 WAS GIVEN RWY HDG FOR TKOF AND TWR FORGOT TO GIVE THEM AN ASSIGNED HDG OF 340 DEGS AFTER TKOF. CONTRIBUTING FACTORS: ACFT ON 2 PARALLEL RWYS ARE SIMULTANEOUSLY GIVEN TKOF CLRNC ON SEPARATE FREQS WITHOUT BEING TOLD OF THE POTENTIAL CONFLICT. COMBINED WITH NO PUBLISHED SEPARATION PROC (I.E.; DIVERGING TURNS FROM THE 2 RWYS) IT MAKES THE CHAIN OF EVENTS LEADING TO THIS INCIDENT RELATIVELY SHORT AND EASY TO OCCUR. RECOMMENDED CORRECTIVE ACTION: STAGGER TKOFS AND INSURE THE SECOND ACFT HAS THE FIRST IN SIGHT PRIOR TO CLRNC. ESTABLISH PUBLISHED TURNS FOR THE PARALLEL RWYS THAT ENSURE POSITIVE ACFT SEPARATION ON TKOF AND DEP IN CASE THE TWR FAILS TO ASSIGN A HDG.