Parking in Fethiye is very difficult. Even in the evenings.
I generally do what Liz 101 suggested, and use the pazar car park, as long as it isn't Tuesday or Friday.
Not of caution: don't leave your car in the pazar car park after 5pm on a Monday afternoon. I came back from a boat trip to find my car covered in t-shirts and other clothing, as the stall holders set up their stalls in preparation for the following day ...