A BOEING FLC RECEIVED A TCASII RA ON DEP FROM LGA FROM A TRAILING AIRBUS A320 WHICH WAS IGNORED.

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DURING DEP FROM LGA RWY 13 MASPETH CLB; WE GOT AN RA FROM THE TCASII. IF WE HAD TRIED TO FOLLOW THE RA; I BELIEVE IT WOULD HAVE MADE A COLLISION MORE LIKELY. AFTER A NORMAL TKOF; WE REACHED 3000 FT AND STARTED OUR TURN FROM 175 DEG HDG TO 340 DEG HDG AT ABOUT 5 DME. IMMEDIATELY AFTER WE STARTED OUR TURN; DEP CALLED AND ASKED IF WE HAD STARTED OUR TURN YET. WE REPLIED AFFIRMATIVE AND DEP CALLED ACFT Y TO TELL HIM HE WAS TURNING INSIDE OF US. (AN ACFT Y A320 DEPARTED AFTER WE DID ALSO FROM RWY 13 ON MASPETH CLB.) DEP TOLD US TO EXPEDITE OUR CLB THROUGH 6000 FT. AFTER WE ACKNOWLEDGED HE TOLD ACFT Y TO STOP HIS CLB AT 5000 FT. WE WERE ALREADY PASSING 5600 FT AND CLBING ABOUT 2000 FPM. AT THAT TIME WE GOT A TCASII RA TO 'DSND' AND THE RED ARC ON TCASII SHOWED WE NEEDED 2500 FPM DOWN. I LOOKED OUT AND SAW THE ACFT Y A320 AT ABOUT OUR 4 O'CLOCK POS; BELOW US; ON AN INTERCEPT HDG. WE DISREGARDED THE TCASII; KEPT ACFT Y IN SIGHT UNTIL HE CROSSED DIRECTLY BELOW US. IF WE HAD NOT HAD HIM INSIGHT AND HAD TRIED TO FOLLOW THE RA I BELIEVE WE MAY HAVE HIT THE A320. WE DIDN'T FIND OUT IF THE ACFT Y A320 GOT AN RA. ATC NOTICED AND CORRECTED THE CONFLICT BUT THE TCASII RA WORRIES ME.