AN ACR LTT FLC; ASSIGNED RWY 24L; WAS TOLD TO REMAIN TO THE L OF AN ACR A320 THAT WAS ASSIGNED RWY 24R. THE A320 FLC OVERSHOT THEIR TURN TO FINAL APCH COURSE AND LINED UP ON RWY 25R AND THE LTT FLC LINED UP WITH RWY 25L. APPARENTLY; THERE WAS NO CONFLICT.

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ON DSCNT TO LAX; ATIS RPTED 600 FT SCATTERED WITH VISUAL APCHS TO ALL RWYS. ON CHK IN WITH THE FINAL CTLR OVER DOWNTOWN LA IT WAS OBVIOUS THAT A MARINE LAYER EXTENDED TO THE EASTERN BOUNDARY OF THE ARPT; AND WE DIDN'T RPT THE ARPT IN SIGHT. THE CTLR ASKED IF AN ACR A320 WAS IN SIGHT AT 2 O'CLOCK POS AND APPROX 6 MI (JUST E OF THE SMO VOR ON DOWNWIND). WE RPTED 'YES' AND WERE INSTRUCTED TO FLY A HDG OF 210 DEGS AND CLRED FOR A VISUAL APCH TO RWY 24L TO FOLLOW THE A320 THAT WAS GOING TO RWY 24R. WE STILL DIDN'T HAVE THE ARPT IN SIGHT; AND WE ASSUMED THAT THE A320 HAD RPTED IT IN SIGHT FROM THEIR POS ON DOWNWIND. IN OUR TURN FROM 140 DEGS TO 210 DEGS WE NOTICED THAT THE A320 WAS QUITE HIGH; AND HAD DELAYED THEIR BASE LEG TURN. WE HAD TO GET ON THE OTHER SIDE OF THE A320; AND IT WAS OBVIOUS THAT A HDG OF 210 DEGS WOULD HAVE TAKEN US TOO CLOSE TO THE A320. I TOLD THE COPLT TO ROLL OUT ON A HDG OF 180 DEGS TO IMPROVE OUR SPACING ON THE AIRBUS. AT THAT TIME WE STILL DIDN'T SEE THE RWY; WE STILL HAD TO GET ON THE OTHER SIDE OF THE NOW CONVERGING A320 (ON THE BASE LEG) AND WE HAD A 65 DEG INTERCEPT TO THE FINAL APCH COURSE. WITH ALL OF OUR ATTN OUTSIDE TO FIND THE RWY AND MAINTAIN SPACING ON THE A320; NEITHER ONE OF US NOTICED THE PASSING OF THE LOC FOR RWY 24L. ABOUT THE TIME THE A320 ROLLED OUT ON FINAL (AND WAS NOW DOING S-TURNS TO THE N TO LOSE ALT) WE NOTICED THE APCH END ONLY OF 2 RWYS; THE A320 WAS TRYING TO LINE UP FOR THE R ONE AND WE LINED UP FOR THE L ONE. IT WAS ALMOST COMPLETELY DARK AND THE ONLY PART OF THE ARPT VISIBLE WAS THE APCH ENDS OF THE 2 RWYS WE WERE HEADING FOR. SOCAL TOLD US TO SWITCH TO TWR; AND UPON DOING SO WE HEARD THE A320 BEING INFORMED THEY WERE LINED UP FOR RWY 25R; NOT RWY 24R; AND THEY WERE CLRED TO LAND ON RWY 25R. ON CHK IN WE WERE INFORMED WE WERE LINED UP FOR RWY 25L; NOT RWY 24L; AND WE WERE CLRED TO LAND RWY 25L. ALTHOUGH IT WAS LEGAL FOR ATC TO CLR US FOR A VISUAL WITH ONLY THE PRECEDING ACFT IN SIGHT; THE CONDITIONS WERE NOT 600 FT SCATTERED AS RPTED; BUT CLOSER TO 600 FT BROKEN; AND ALMOST OVCST; WHICH WAS LESS THAN THE 1000 FT CEILING LEGALLY REQUIRED. THE LACK OF LIGHTS WAS ALSO A FACTOR. ALSO A FACTOR WAS OUR REQUIREMENT TO MAINTAIN ADEQUATE SPACING ON THE A320; THE CONCERN FOR WAKE TURB AVOIDANCE; OUR FORCED ATTN OUTSIDE THE COCKPIT; AND THE BELIEF THAT WE HAD TO BE ON THE OTHER SIDE OF AN AIRPLANE THAT WAS GOING TO THE WRONG RWY.