B737-300 CREW WAS GIVEN A HEADING AFTER TKOF THAT TURNED THE ACFT INTO A DEPARTING MD80 TAKING OFF ON A PARALLEL RWY AT DFW.

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WE TOOK OFF FROM DFW ON RWY 18L (W SIDE OF FIELD; WITH W TWR FREQ 124.15). OUR INSTRUCTIONS PRIOR TO TKOF WERE TO FLY HEADING 155 DEGS AFTER TKOF (A 19 DEG L TURN FROM RWY HEADING). CLRED FOR TKOF. AT 400 FT AFE; I BEGAN A SHALLOW L TURN TO 155 DEGS. JUST AS I ROLLED OUT ON 155 DEGS; TWR CTLR CAME ON WITH A SENSE OF URGENCY DIRECTING US TO TURN R TO 190 DEGS. WE DID SO AND THEN NOTED BOTH VISUALLY AND ON TCASII; AN MD80 WHO HAD TAKEN OFF FROM THE E SIDE OF THE FIELD (DIFFERENT TWR FREQ). IT WOULD APPEAR THAT THE 2 TWR CTLRS DID NOT COORDINATE THEIR ACTIONS WELL AS OUR CTLR HAD GIVEN US A HEADING WHICH WAS GOING TO CONSTANTLY REDUCE THE LATERAL SEPARATION BTWN US. NO TCASII WARNING OF ANY KIND (TA OR RA) OCCURRED. IN LARGE PART; THIS PROB WAS CTLR INDUCED. I DID HAVE TO ASK MYSELF IF THERE WAS ANYTHING I COULD HAVE DONE TO PREVENT THE CTLR'S ERROR FROM GOING AS FAR AS IT DID. I BELIEVE IF I HAD ASKED THE CAPT IF WE WERE CLR L BEFORE I STARTED MY TURN AT 400 FT AFE; THAT THE CAPT WOULD HAVE SEEN THE MD80 AND QUESTIONED THE CTLR BEFORE TURNING. ANOTHER LESSON LEARNED.