A B737-300 PARKED AT GATE HAD ZERO GND SPOILER DAMAGED BY DUAL JETBRIDGE. SPOILERS WERE EXTENDED WHEN STRUCK BY JETBRIDGE.

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AFTER LNDG ON RWY 35R AT ZZZ; FORGOT TO LOWER (STOW) SPD BRAKES. DURING CREW DEBRIEF IN JETWAY; THE NEW CAPT WAS BECKONED BY THE OPS AGENT TO LOOK AT THE COMPUTER SCREEN INSTALLED BY THE JETWAY CTL PANEL. APPARENTLY; A WARNING OR SOME OTHER TYPE MESSAGE WAS BEING DISPLAYED ON THE SCREEN. THE NEW FO CAME UP JETWAY STAIR AFTER HIS WALKAROUND INSPECTION AND MENTIONED THAT THE AFT JETWAY BRIDGE MAY HAVE HIT AIRPLANE. I WENT TO JETWAY WINDOW TO LOOK OUTSIDE AND SAW THAT THE OUTBOARD CORNER OF THE L OUTBOARD GND SPOILERS WAS BENT FORWARD. LOOKED LIKE A 2 INCH TRIANGULAR AREA WAS BENT FORWARD. I ENTERED DAMAGE INFO INTO LOGBOOK AND NOTIFIED MAINT CTL. BECAUSE THE CORNER OF SPOILER WAS FOLDED FORWARD TOWARD THE FRONT OF THE ACFT; I BELIEVE THE DAMAGE HAPPENED WHEN THE AFT JETBRIDGE WAS MOVED INTO POS. ASSISTANT CHIEF PLT NOTIFIED OF DAMAGE. DAMAGE COULD HAVE BEEN AVOIDED HAD I LOWERED THE SPD BRAKE; PART OF THE NORMAL FLOW AFTER CLRING THE RWY. NO EXCUSE FOR NOT LOWERING THE SPD BRAKES. SUPPLEMENTAL INFO FROM ACN 577415: UPON ARRIVING AT THE GATE; THE 'PARKING CHKLIST WAS COMPLETE AND BOTH CREW MEMBERS PACKED UP FOR AN ACFT CHANGE. WHILE IN THE JETWAY; THE FO PERFORMING A WALKAROUND CAME UP TO INFORM US THAT THE REAR ARM OF THE DUAL JETBRIDGE HAD STRUCK AND BENT THE L OUTBOARD SPOILER. WE LOOKED OUT THE WINDOW OF THE JETBRIDGE AND NOTICED THE SPOILERS WERE STILL DEPLOYED. SINCE THE SPOILERS WERE IN THE 'UP' POS; THE JETBRIDGE ARM MADE CONTACT TO THE SPOILER WHILE THE JETBRIDGE WAS MOVING INTO POS FOR THE REAR ENTRANCE/EXIT OF THE ACFT. IT APPEARS THE CAPT FORGOT TO LOWER THE SPD BRAKES DURING TAXI IN AND THE FO FORGOT TO DOUBLECHK THAT THE SPD BRAKE LEVER WAS IN THE CORRECT POS. CONTRIBUTING FACTORS COULD BE THE 'GATE SVCS' PROC AT ZZZ. THE FO MADE A COMMENT ABOUT REMEMBERING NOT TO START THE APU DURING TAXI IN. DURING THE DISCUSSION; BOTH PLTS AFTER LNDG FLOW GOT DISRUPTED AND THE SPD BRAKE LEVER WAS NEVER LOWERED. CORRECTIVE ACTIONS WOULD BE TO JUST DOUBLECHK FLAP AND SPD BRAKE LEVER POS WHILE TAXIING IN.