The clicking is likely the starter mounted solenoid closing to the key signal and then popping back open due to the resulting low voltage. Throw the parallel switch to bring other batteries into the start circuit and see what happens. If this works you know the starter is OK and now back to figuring out why one, new, battery wont’ turn your motor over as it should be well capable of doing which means the load test mentioned earlier, likely where you bought it will do this for you at no charge to establish if the battery needs to be warrantied.
If it tests OK charge it back up as the load test puts a hurt on batteries and then you’re chasing the bad ground as someone here said earlier.
Check and freshen up all the grounds, engine ground, this will freshen up the starter ground as it's grounded to the engine, battery grounds, all of them, Dissemble, sand with Emery cloth, reassemble and cote with dielectric grease. Blade and plug connectors can be freshened up by disconnecting and connecting them a few times, throw a little grease in them too.
Folks here talk about adding to or replacing the main ground line to the engine. I did this on both my R25 and R29 and the electrical system seemed more lively afterward although I can't prove it.
