Intermittant no start?

[GulfcoastSD]

My dad has been having an issue with his stock 2002 F350. Specifically it will not start some times until several on/off cycles of the key. When it does not want to start the starters does not seem to even try to engage. I can hear what I am thinking is the vacuum pump above the passenger wheel well come on and a slight clicking noise when he tries to start it. The problem is intermittant an has been going on for the last month or so. He has cleaned the terminals and battery connections and is replacing the starter today even though the guy at the parts house tested his starter and said it was good. Any other ideas?

[GulfcoastSD]

New starter but same issue

[NoRalPh]

I'd be messin' with the Shift Selector for an out-of-adjustment Neutral Safety Sw if it was an Automatic or the Clutch Sw if a Stick...