MU 300 FLT CREW DEPARTED WITHOUT TKOF CLRNC.

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THE PROB AROSE PRIOR TO DEPARTING DAL; ON THE SECOND LEG OF THE DAY; AND FINISHING THE BEFORE TKOF CHKLIST WITH A NEW HIRE COPLT IN TRAINING. WHILE HOLDING SHORT AND LISTENING TO A BUSY CTL TWR (GIVING THE TIME OF DAY); I GAVE A FINAL REVIEW OF THE DALLAS 8 DEP PROC AND REVIEWED TKOF ITEMS WITH THE NEW COPLT. I TOLD THE COPLT TO LET TWR KNOW THAT WE WERE READY FOR DEP. TWR TOLD US TO HOLD SHORT. AFTER HOLDING SHORT; TWR INSTRUCTED OUR ACFT TO TAXI INTO POS AND HOLD RWY 13R; WITHOUT AN EXPLANATION. WHILE I TAXIED INTO POS TO HOLD; I WAS HEARING ALL THE OTHER CLRNCS AND INSTRUCTIONS FROM TWR AND HEARD THE TWR SAY 'CLR FOR TKOF.' I SAID TO THE COPLT 'CLR FOR TKOF;' NOT KNOWING THAT THERE WAS ANOTHER ACFT ON THE PARALLEL RWY (RWY 13L); BECAUSE OF NOT BEING ABLE TO SEE THE OTHER RWY FROM RWY 13R. I PROCEEDED TO TAKE OFF. AFTER TKOF AND CLBING OUT; I NOTICED A LEARJET 45 CLBING AHEAD OF ME; MUCH FASTER; AT MY 10:30 - 11 O'CLOCK POS; OFF THE PARALLEL RWY; 13L. IN THE CLB; TWR STARTED TO GET AN INSTRUCTION TO US ABOUT TFC; BUT STOPPED TALKING IN THE MIDDLE OF THE INSTRUCTION SENTENCE. TWR CAME BACK INSTRUCTING US TO 'TURN; TURN R 15 DEGS.' BEFORE; DURING AND AFTER THE TIME OF THE CLRNC TO 'TURN;' WHILE IN THE CLB; I HAD A VISUAL ON THE LEAR 45. I PROCEEDED WITH THE CLRNC TO 'TURN' AND STARTED LOOKING FOR OTHER TFC. ON A HDG OF 145 DEGS; TWR INSTRUCTED US TO CONTACT REGIONAL APCH CTL. MY COPLT CHKED IN WITH THE ALT WE WERE OUT OF; AND CLBING TO; AND HDG. I CALLED BACK TO APCH AND SAID 'WHAT WAS ALL THAT ABOUT?' THE APCH CTLR REPLIED CALMLY; 'I DON'T KNOW? BUT WE'LL GET YOU ON YOUR WAY.' JUST BEFORE APCH HANDED US OVER TO ZFW; HE INSTRUCTED US TO CALL LOVE TWR. AT OUR DEST; I CALLED LOVE TWR. THE GENTLEMEN SAID 'THANKS FOR CALLING BACK.' AT THAT POINT HE INFORMED ME THAT I TOOK OFF RWY 13R AT LOVE FIELD WITHOUT A TKOF CLRNC. SOME CONTRIBUTING FACTORS WERE 1) TRAINING A NEW HIRE COPLT WITH THE COMPANY'S SOP'S; 2) FREQ CONGESTION; 3) THE ABILITY TO VISUALLY CHK THE PARALLEL FOR DEPARTING TFC; 4) NOT VERIFYING CLRNC INSTRUCTIONS FOR TKOF; 5) AND BEING COMPLACENT IN FLYING IN AND OUT OF DALLAS LOVE FIELD WKLY.