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Hello. Welcome to Just Answer. Please allow me to assist you. This sounds to me like the shift cable attached to the shift lever on the transaxle is worn, stretched or otherwise faulty, or, the physical attach point where the cable meets the shift lever is faulty. When any of the above occur, the shifter indicates "park", but the shift LEVER on the transaxle is not fully inserted and held in the park position. Further visual diagnosis would be needed here, but I am reasonable sure you will find the cause of the issue where the shift cable meets the shift lever on the transaxle.
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Indeed, a Lexus dealership can rectify this. I would not be inclined to think that Lexus will cover this repair; the vehicle is 12 years old.
If it needs a shift cable, the price should be in the 250-350 dollar range; parts and labor.