PRIVATE PLT OF AN SMA SEL ACFT LANDED ON THE WRONG PARALLEL RWY DUE TO A MISUNDERSTANDING BTWN THE PLT AND THE CTLR.

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WE WERE IN R HAND PATTERN AFTER TOUCH-AND-GO LNDG ON RWY 17R. ON ENTERING DOWNWIND LEG THE PIC; FLYING FROM L SEAT; REQUESTED CLRNC TO LAND FULL STOP. CTLR APPROVED LNDG. BEFORE TURNING ONTO BASE LEG; TWR ADVISED THAT WE WERE CLRED TO LAND #2 AFTER ACFT ON SHORT FINAL. PIC ACKNOWLEDGED. (AT THIS TIME PIC INDICATED TO ME HE HAD ACFT ON SHORT FINAL.) AS PIC TURNED SHORT FINAL AND WAS LINING UP FOR RWY 17L; TWR CALLED UNKNOWN ACFT QUESTIONING WHETHER IT WAS LNDG ON RWY 17L. TWR THEN TALKED TO A SECOND ACFT APPARENTLY ON LONGER FINAL APCH TO RWY 17L TO SAY AN ACFT CUT THEM OFF AND RECOMMENDING THEY USE S-TURNS FOR SPACING. THAT PLT ASKED APPROVAL TO GAR. AFTER WE LANDED; TWR ADVISED PIC THAT HE WAS APPROVED FOR RWY 17R. THE PIC STATED THAT HE UNDERSTOOD HE WAS APPROVED TO LAND ON RWY 17L. THE CONFUSION CAME ABOUT BECAUSE THE TWR CTLR DID NOT SPECIFICALLY STATE WHICH PARALLEL RWY THAT WE WERE CLRED FOR AND PIC DID NOT QUESTION THE TWR SINCE HE UNDERSTOOD HE WAS CLRED FOR FULL STOP LNDG ON THE LONG RWY; RWY 17L. AS PAX; I WAS NOT CLEAR ABOUT WHICH RWY WE WERE CLRED TO LAND ON. THE SECOND ACFT APPARENTLY WAS ABLE TO SEE A CONFLICT ARISING AND GAR.