What you are referring too is the regular brake, there is no separate parking brake, just the pedal lock mechanism to hold the brakes once applied.
Like I mentioned, try tightening the center nut just a little and see if that helps the brake function. It would help to jack up the rear of the mower if possible.
These differential units' w/brakes were used in many rider models besides Murray back then, so maybe used brake pads can be located on a salvage unit. Brake pads were not a very common replacement item, so most riders broke down for other reasons before the brakes ever got worn out.
Another idea is to disassemble carefully and see if you can fabricate some thin sheet metal shims that will fit in place behind each pad. It does not take anything real thick to shim them.
A little history:
Murray was an independent company that built riders until they filed for bankruptcy in 2004, then B&S/Generac/Gentek bought them out. That makes older replacement parts a little hard to find besides "generic" belts, pulleys, spindles and blades.