I was always under the impression that leash laws were city ordinances and not state laws.
I found this online:
Several states have state-wide dog leash laws and requirements. These laws are usually referred to as "Running at Large Statutes." In states without "Running at Large Statutes," local governments, such as counties, towns, cities, municipalities, and boroughs, often enact their own leash laws.
But even where there are state-wide leash laws, it's often the case that the state government permits local governments to pass their own leash laws. Often, the local laws are stricter than the state laws. If any confusion arises over the leash laws in your home, contact your local government for more information
Missouri has a "State Lands Leash Law." Under this law, dogs must be on leashes no longer than 10 feet when they are in state parks or on state historic sites.
Additionally, Missouri law prohibits dogs that have rabies, or dogs that have been exposed, to rabies to run at large.