SO just to be clear, the brake works and stops OK but the parking mechanism does not hold so the brake dissengages and the mower go rolling down the hill ( to fetch a pail of water ? )
Behind the fuel tank is the control rod that goes from the dash lever to the brake under the floor.
Down the bottom inside the dash is a small spring which holds the control rod up keeping the brake on
You should be able to get at it at worst you will need to remove the tank.
take the deck off then get under the mower and you will see how it works, dirt simple.
A twig or some long grass underneath underneath the mower wll prevent the parts from meshing properly.