The Fly Classic was one of Parker's original Fly models which featured a mahogany body compared to other base tonewoods elsewhere in the Fly line. After Ken Parker sold the company in 2003, and production was relocated from Massachusetts to Illinois, the new ownership instituted a number of cost-saving production changes which were debuted under the "Refined" banner at Summer NAMM.

Parker Fly Classic models can be identified by the letter suffix "BMH" which stands for basswood neck, mahogany body. This code is found at the end of serial number on pre-refined models, and elsewhere on the body for refined models (typically the back of the headstock).