Did anyone check the ignition switch itself? (The place where the key goes.)
Are you using a replacement key? Perhaps the chip in the key is bad. Or the device that reads the chip is failing.
Have you pulled the starter and had it checked? There could be a bad spot in the wiring.
If these are not the problem, my diagnosis is that you have an intermittent short or loose connection somewhere in the system. Could be the battery cables. Could be any wire.
I can't see how the parking brake is involved...unless pushing it moves a wire that is in the process of shorting out.
The clicking noise suggests that the starter is either not getting juice from the battery or the alternator/regulator is not allowing the battery to charge.
Have you done an internet search on this problem? Check Technical Service Bulletins for your car. And get a new mechanic.