PA28 STUDENT PLT AND INSTRUCTOR PLT LANDED ON THE WRONG RWY.

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WE STARTED OUT OUR LESSON IN THE PATTERN; SO WE WOULD KNOW EXACTLY HOW MUCH TIME WE HAD TO WORK ON MANEUVERS IN THE PRACTICE AREA. WE TOOK OFF FROM RWY XXL AND WERE INSTRUCTED TO MAKE L TFC FOR XXR (PARALLEL RWYS). I WAS FLYING THE ACFT. I REPEATED THE INFO BACK CORRECTLY AND PROCEEDED WITH CONFIGURING THE ACFT FOR A TOUCH AND GO LNDG. AT NO POINT IN TIME WAS THERE CONFUSION ABOUT THE SIT; OR WHAT I WAS TO DO. I CONTINUED ON BASE AND TURNED FINAL; EVERYTHING APPARENTLY NORMAL. I THEN PERFORMED MY TOUCH AND GO ON RWY XXL; INSTEAD OF RWY XXR; PER INSTRUCTIONS. NEITHER MYSELF NOR MY INSTRUCTOR NOTICED ANYTHING WAS WRONG; THE PATTERN WAS FORTUNATELY EMPTY. TWR THEN CONFIRMED THE RWYS WITH US AND WE CONFIRMED (TO OUR HORROR AND SHOCK) THAT WE HAD INDEED LANDED ON THE INCORRECT RWY. THE REST OF THE FLT WENT THANKFULLY; UNEVENTFUL. SUPPLEMENTAL INFO FROM ACN 613836: MY STUDENT AND I WERE CLRED FOR RWY XXR AND LANDED ON RWY XXL WITHOUT REALIZING. THERE WAS A VEHICLE ON THE RWY THAT WE DID NOT SEE; HOWEVER; THERE WAS NO INCURSION. THE TWR POINTED EVERYTHING OUT TO US AND CLRED US FOR ANOTHER TOUCH AND GO ON RWY XXR AGAIN. WE PROCEEDED THE REST OF THE FLT; UNEVENTFULLY AND LANDED.