I will need to confirm, but I have a manual transmission, as I recall if I try to start with the clutch out the deck turns off but no turn over. So it could still be the transmission park switch.
If everything else works you can verify the starter electrical path by taking the starter out hooking the wiring back up then try the key. Obviously this is a two man job and you want the starter well supported to avoid damaging the wiring from a fall. If she spins then your electrical "should' be good, if not you are back to checking electrical components. Clicking is generally poor battery, failed starter, failed relay, failed solenoid or serious engine issues. You have taken care of 2 out of three, as you have mentioned no issues with the motor it should be electrical.