ACR ISSUED A L TURN AFTER TKOF NOTICES ANOTHER ACFT OFF THE PARALLEL RWY. ATC ISSUED RWY HDG CLRNC.

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FLYING FLT OUT OF DAYTON HOLDING SHORT 24R. AN ACR WAS ON FINAL. WE WERE WAITING FOR WT AND BAL. RECEIVING OUR WT/BAL WE NOTIFIED TWR WE WERE READY. DAYTON TWR CLRED US INTO POS AND HOLD. WE RESPONDED AND TOOK POS. TWR THEN CLRED US FOR AN IMMEDIATE TKOF. WE RESPONDED AND STARTED TKOF. WE WERE GIVEN 5000 FT AND L TO 220 DEGS. UPON LIFT OFF I NOTICED A NW COMMUTER HAD JUST LIFTED OFF 24L. I QUESTIONED TWR ABOUT OUR CLRNC. A LONG SILENCE. TWR TOLD US TO HOLD RWY HDG; MAINTAIN 5000 FT. TFC WAS IN SIGHT. THERE WAS NEVER A TFC CONFLICT. WE WERE SENT OVER TO DEP. AT THAT TIME; I ASKED DEP TO CALL TWR AND SEE IF THEY HAD A PROBLEM WITH OUR FLT ON TKOF. DEP RESPONDED NO. WE PROCEEDED ON TO CLT. SUPPLEMENTAL INFO FROM ACN 200714: I SWITCHED OFF THE LAMPREY RECEIVER AS THE CAPT WAS RESPONDING TO A TKOF CLRNC OF CLRED FOR IMMEDIATE TKOF TURN L HDG 220 DEGS. WE TOOK OFF AND STARTED A L TURN. THE TWR SEEMED SURPRISED THAT WE WERE AIRBORNE AND GAVE US A CLRNC TO MAINTAIN RWY HDG; THAT THERE WAS TFC THAT DEPARTED ON A PARALLEL RWY.