Clicking solenoid is a sign that it is not getting enough power
Around 9 to 10 V will trip the solenoid but when the connection is made for the starter the starter draws a lot of amps which will cause the voltage to drop
When the voltage drops to below the 9/10 volts the solenoid turns off, the high current draw stops the available voltage rises and the solenoid closes again, the high current draw drops the voltage which gets too low the solenoid turns off, the current load dissapears so the voltage rises .
You hear the machine gun clicking
SO the actual problem is the starter can not get enough amps or wants more amps than the mower can supply .
The most common cause of this is a bad ground strap
The test is to run a ground jumper from the battery - to a good ground near the starter
Problem goes away confirms this.
Note on ZTR's I have seen a few where water has wicked into battery cables and rotted through the wires under the insulation so they look good, will Ω test good but not carry any current
Can happen to either or both cables.
Assuming that the ground jumper did not make the starter work then the next trick is to run a jumper from the + terminal on the starter motor to the + terminal of the battery ( connect the starter first) . The starter should spin if it and the battery is good .
Assuming no joy here either then jump from a known good battery as above directly to the starter
If the starter spins the mower battery is bad
If the starter does not spin then it is faulty.