IFR C340 DESCRIBED CONFLICT WHEN ASSIGNED 360 OFF RWY 9R AND IFR TFC OFF RWY 9L HEADING 060; ATC ALLEGED VISUAL SEPARATION; NO OPERROR.

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I WAS FLYING THE C340/A AND DEPARTING RWY 9R AT FLL. TWR CLRED ME FOR TKOF ON RWY 9R WITH A TURN TO HDG OF 360 DEGS. (RWY 9L TWR IS A DIFFERENT FREQ SO I WAS NOT TALKING TO THEM.) I DEPARTED RWY 9R; CLBED 400 FT AGL; AND TURNED TO A HDG OF 360 DEGS. THIS PUT ME RIGHT OVER THE END OF RWY 9L AT 500 FT MSL. I LOOKED L AS MY TCAS ALARM WENT OFF; TO SEE A BE1900 LOOMING IN VIEW; CLBING RIGHT INTO ME. I BANKED HARD R AND PULLED UP AS THE OTHER ACFT DISAPPEARED BENEATH ME. BEING A SMALLER MUCH SLOWER ACFT; THERE REALLY WAS LITTLE I COULD DO TO AVOID BEING HIT. I AM FORTUNATE THAT HE WAS A FEW SECONDS BEHIND ME; OR ELSE.... I ASKED FOR THE PHONE NUMBER OF THE CTL TWR. THE BE1900 ALSO CALLED ME IN THE AIR TO SAY THAT HE HAD BEEN GIVEN A HDG UPON TKOF OF 060 DEGS. I HAD BEEN GIVEN A HDG OF 360 DEGS. 2 ACFT DEPARTING PARALLEL RWYS AND GIVEN INTERSECTING HDGS; BOTH ON IFR FLT PLANS; TALKING TO DIFFERENT TWR CTLRS CANNOT BE PROPER OP. I FEEL THAT ATC IS CLRLY AT FAULT. THE TWR RELAYED A MESSAGE TO ME LATER IN THAT SAME FLT ASKING THAT I CALL THEM ONCE ON THE GND. THE SUPVR APOLOGIZED FOR THE NMAC; BUT SAID THAT BOTH CTLRS HAD BOTH ACFT IN SIGHT. I BELIEVE A CHANGE NEEDS TO TAKE PLACE ON TKOF PROCS AT THE ATC LEVEL FOR DEPARTING ACFT AT FLL. OTHERWISE; THERE WILL BE MIDAIR COLLISIONS WHICH WILL INEVITABLY INVOLVE EVEN MORE CASUALTIES ON THE GND IN THIS HEAVILY POPULATED URBAN CITY.