AFTER LNDG PSP; A WW24 CREW MAY HAVE CROSSED THE PARALLEL RWY WITHOUT CLRNC.

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AFTER LNDG ON RWY 31L AT PSP; WHICH INVOLVED MODERATE TURB AS WELL AS A 10-15 KT LOSS OF AIRSPD OVER THE NUMBERS DUE TO WINDSHEAR; WIND WAS RPTED AS APPROX 290 DEGS AT 18 KTS GUSTING TO 25 KTS. DURING ROLLOUT; TWR CTLR TOLD US TO TAKE THE NEXT L OR R TURN. I ALSO BELIEVE THE CTLR SAID 'TAXI TO THE RAMP.' AS I EXITED TO THE R; ON TXWY J; THE CTLR ASKED US 'HOW WAS YOUR RIDE?' THE CO-PLT REPLIED 'WE LOST 10-15 KTS OVER THE NUMBERS.' THE CTLR REPLIED 'THANKS; ACR Y; DID YOU COPY THAT RPT?' HE REPLIED; 'YES; AND THANKS FOR THE RPT.' DURING THAT EXCHANGE; I CONTINUED TAXIING. RWY 31L&R ARE FAIRLY CLOSE TOGETHER; THEREFORE; AS I WAS APCHING RWY 31R; I VERIFIED WITH MY CO-PLT THAT WE WERE CLRED TO CROSS. HE REPLIED; 'CTLR CLRED US TO THE RAMP.' BELIEVING I HAD HEARD THE SAME; I CONTINUED TO CROSS. UPON REACHING THE OTHER SIDE; I HEARD THE TWR TELL ANOTHER ACFT TO CANCEL TKOF CLRNC. THERE WAS A PAUSE AND THEN THE TWR ASKED THE PLT IF WE WERE ACROSS AND HE REPLIED 'YES.' THE TWR THEN SAID THAT WE WERE NOT CLRED TO CROSS. WE WERE NEVER GIVEN HOLD SHORT INSTRUCTIONS OR TOLD TO CONTACT GND. I CONTINUED MY TAXI TO THE RAMP. UPON REACHING THE RAMP; WE WERE TOLD TO CONTACT GND. I CONTACTED GND AT THAT TIME AND WAS TOLD TO CONTACT THE TWR VIA THE TELEPHONE. CONTRIBUTING FACTORS: 1) STRESSFUL LNDG DUE TO MODERATE TURB AND WINDSHEAR. 2) NONSTANDARD COM BTWN TWR AND CREW DURING CRITICAL PHASE OF OPS. 3) NOT CLARIFYING TAXI INSTRUCTIONS BY CREW OR TWR. 4) CREW UNFAMILIARITY (FLYING WITH CONTRACT PLT). 5) FATIGUE DUE TO 4 HR 15 MIN FLT AS WELL AS 9 HRS DUTY TIME. CORRECTIVE ACTIONS: 1) USING STANDARD COMS. 2) CLARIFY ALL INSTRUCTIONS OR LACK OF INSTRUCTIONS. 3) BE MORE ACCURATE DURING CRITICAL PHASES OF OPS. SUPPLEMENTAL INFO FROM ACN 612851: I WAS ACTING AS CO-PLT WITH A CAPT I RARELY FLY WITH. AFTER TOUCHDOWN DURING ROLLOUT; I BELIEVE TWR SAID L OR R TURN TO THE RAMP. WE NOTICED A KING AIR AT THE OTHER END OF RWY 31R IN POS; BUT WAS NOT ROLLING. WE CLRED RWY 31R AND THE KING AIR DEPARTED. TWR ASKED THE KING AIR IF WE WERE CLR OF RWY 31R WHEN HE DEPARTED. HE SAID YES. WE DO NOT RECALL BEING TOLD TO HOLD SHORT OF RWY 31R. TOO MUCH CHATTER ON THE RADIO DURING LNDG AND CREW'S LACK OF CLARIFYING TAXI INSTRUCTIONS. CREW SHOULD LISTEN CLOSER TO TAXI INSTRUCTIONS AND REQUEST CLARIFICATION WHEN NOT 100% SURE.