A B737 PLT FAILED TO SET THE PARKING BRAKE DURING ENG START AND THE ACFT ROLLED INTO THE TOWBAR AFTER PUSHBACK.

Narrative

WHEN CHKING THE ACFT LOGBOOK; I NOTED AN ENTRY MADE THAT THIS APU PUT OUT LOW AIR AND HAD A DELAYED START. WE ALSO NOTICED THAT AS FAR AS AIR CONDITIONING THE ACFT; THE APU WAS DOING A POOR JOB. I PRE-BRIEFED THE FO THAT WE WOULD TURN THE ENG HYD PUMPS OFF AND GET THE BEST N2 BEFORE WE START. (IN MY IRREGULARITY RPT; I NOTED THE ACFT LOG PAGE NUMBER) PUSHBACK WAS NORMAL AND WE WERE CLRED TO START. I ELECTED TO TURN ENG #2 FIRST SO WE COULD GET AIR TO THE PAX SOONER AS IT WAS A HOT DAY. WE COULD ONLY GET 22% N2 BUT CONTINUED THE START. JUST AS ENG #2 N2 REACHED 46%; THE TUG OPERATOR CALLED FOR BRAKES SET. I SET THE BRAKES; CALLED BRAKES SET AND THEN CALLED FOR ISOLATION VALVE CLOSE AND START #1. WHAT I DID NOT DO WAS TO VERIFY THE BRAKE RED LIGHT ON; BUT INSTEAD WAS WATCHING THE EGT ON ENG #2. MY OUTSIDE VISION JUST STARTED TO NOTE MOVEMENT WHEN I HEARD THE NOSEWHEEL MAKING CONTACT WITH THE TOWBAR. I IMMEDIATELY JUMPED ON THE BRAKES TO STOP THE ACFT; BUT NOT BEFORE RIDING UP ON THE TOWBAR. WE ASKED THE TUG DRIVER IF ANYONE WAS HARMED AND HE SAID NO ONE WAS TOUCHED BUT WE BETTER CALL MAINT. WE DID CALL AND THEY CAME OUT TO DISLODGE THE TOWBAR AND INSPECTED THE NOSE GEAR AND CERTIFIED IT OK AND RELEASED US FOR FLT. THE FLT TO ZZZ1 AND BACK TO ZZZ2 WAS UNEVENTFUL. I HAVE TO ASSUME THAT I NEVER TRULY SET THE BRAKES EVEN THOUGH I KNOW IN MY MIND THEY WERE SET. I AM ALSO EXTREMELY THANKFUL THAT NO ONE WAS HURT.