Despite the fact that it is a new battery and alternator, the battery could have a bad cell and not able to provide enough "grunt" to start the car even though all the lights work. I agree the neutral safety switch could be bad. Another easy thing to try is to simply jump the battery with another car and see what happens......

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2012 Mercedes ML350 Bluetec 102K (hers)
2005 Corvette 55K (fun car)
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1998 Volvo S70 T5 Turbo 211K (kids)
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