After you turn the engine by hand and hit the ignition again, does it spin or does it still just click. If it spins after that, you may need to check your adjustment on the valves. Something else we have experienced is if you get a battery, say from wal-mart, the cranking amps may not be what you need for your mower. The ones we sell for mowers have 330 cca. That is the standard battery though, there are other mowers that will take special batteries.
It is odd that you are getting a click with all those new parts, but if it is clicking as you say we should be able to rule out the safety switches.