ACR HAS TFC CONFLICT WITH TFC FROM PARALLEL RWY DURING TKOF.

Narrative

FLT WAS CLRED INTO POS ON RWY 1R AND ADVISED TO BE READY. AS SOON AS THE FIRST HALF OF A B737 CLRED THE RWY; WE WERE CLRED FOR TKOF. I TURNED CTL OF THE ACFT OVER TO MY FO. THE TKOF PROCEEDED UNEVENTFULLY AND AT 400 FT AGL A L TURN WAS INITIATED IN ACCORDANCE WITH THE RED ROCK 2 SID. AS MY FO BEGAN THE TURN; I GLANCED TO THE L AND OBSERVED A SINGLE ENG CESSNA DIRECTLY TO OUR L ALSO STARTING A L TURN. SIMULTANEOUSLY A TCASII WARNING SOUNDED. MY ACFT WAS CLBING VERY RAPIDLY AND IN A FEW SECONDS WE PASSED ABOVE AND SLIGHTLY TO THE R OF THE CESSNA. EVERYTHING HAPPENED SO QUICKLY THAT NO EVASIVE ACTION WAS TAKEN TO AVOID THE ACFT. UPON CONTACTING LAS DEP CTL; WE WERE ADVISED OF THE CESSNA TFC. I WAS STARTLED AND STILL SOMEWHAT SHAKEN BY THE SUDDEN APPEARANCE OF THIS ACFT AND COMMUNICATED THIS TO DEP CTL WHEN TIME PERMITTED. I WAS GIVEN A TELEPHONE NUMBER AND ASKED IF I DESIRED AN FAA REPRESENTATIVE TO MEET ME IN SJC. I SAID THAT WOULD NOT BE NECESSARY. UPON ARR IN SJC I SPOKE WITH ATC MGR IN LAS. WE DISCUSSED WHAT HAD HAPPENED AND ATC MGR ADMITTED THE SIT COULD HAVE BEEN AVOIDED HAD THE TWR OPERATOR ADVISED US OF THE CESSNA TFC WHICH HAD DEPARTED JUST AHEAD FROM AN INTXN ON RWY 1L. HE ASSURED ME THAT THE INCIDENT WOULD BE DISCUSSED AT THE NEXT SAFETY MEETING.