I've had a look into the story - and what a story of corruption, intrigue, double-crossing and murder! It makes Al Capone look like your friendly neighbourhood Sunday school teacher and took place mainly in 1920s Chicago too, amongst the taxi companies there, with some well-known names like Hertz involved.
But from what I can glean the old Franklin Parmalee company, established in the mid 1800s and making vehicles primarily for the railway system's baggage transfer (see picture below), was taken over by Checker in 1930. Checker itself was taken over by Cord in 1932, and ended up of course as part of General Motors.
But it does seem that any cab badged as a Parmelee was simply a re-badged Checker, and the puzzle photo seems to be a Y8 model from 1936.
So let's award the point and call it solved!