AFTER VERIFYING CLRNC INTO POS TO HOLD; A DUCHESS PLT IS SURPRISED TO FIND A MOONEY ON GAR OVERHEAD.

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I WAS INSTRUCTED TO TAXI TO RWY 9R. MY READBACK FOR RWY 9R WAS NOT ACKNOWLEDGED. I CALLED THE TWR FROM THE HOLD SHORT LINE OF RWY 9R; AND ASKED FOR TKOF CLRNC. TWR INSTRUCTED ME THAT I WAS IN #2 POS. BECAUSE ANOTHER ACFT WAS ON THE OPPOSITE SIDE OF THE RWY; I ASSUMED THAT HE WOULD BE RELEASED FIRST; MAKING ME #2. AFTER ABOUT 1 MIN; I WAS INSTRUCTED TO GET INTO POS AND HOLD. THAT CALL WAS PICKED UP BY ANOTHER DUCHESS ON RWY 9L AND ACKNOWLEDGED. THE INSTRUCTOR IN THE OTHER DUCHESS ASKED FOR VERIFICATION AS TO WHETHER THEY OR I WERE INTENDED. THE RESPONSE WAS THAT MY ACFT WAS INTENDED. I TAXIED INTO POS AND HOLD. AFTER ABOUT 30 SECONDS A MOONEY CALLED IN AND ANNOUNCED THAT HE WOULD GO AROUND AS A DUCHESS WAS IN HIS WAY. DUE TO THE SUN'S POS; I HAD NOT SEEN THE MOONEY BEFORE. BY THE TIME THAT THE MOONEY CLRED OVER ME; THE GEAR AND FLAPS WERE ALREADY RETRACTED. I RECEIVED A TKOF CLRNC FROM RWY 9R WITH INSTRUCTIONS TO CALL THE TWR FROM A TELEPHONE ASAP. IN THE FOLLOWING TELEPHONE CONVERSATION WITH THE GND CTLR; HE TOLD ME THAT HE HAD CLRED ME TO RWY 9L AND HE ACKNOWLEDGED THAT IT WAS POSSIBLE THAT A MISCOM HAD OCCURRED; SINCE THERE WERE 2 DUCHESSES IN PARALLEL (1 ON RWY 9R AND 1 ON RWY 9L). IT WAS ACKNOWLEDGED BY THE TWR THAT I HAD CALLED IN FROM RWY 9R FOR TKOF CLRNC.