There will be a small wire going to the starter solenoid. Disconnect it and see if you get battery voltage from it to ground when the key is turned to start position. If NO then the problem is in the start circuit. Could be ignition switch or any other safety switch OR a corroded connection. Start at the ignition switch. There should be battery voltage at the S terminal when the key is turned to the start position. If YES then you will need to work your way through the switches. clutch, PTO.
Clean the ground connection from battery to mower. If the engine has a fuel cutoff solenoid on the carb does it "click" when you turn the key to ON?