So, I have a perfectly good starter, battery, and alternator. Nevertheless, on random occasions when I turn the key, all the lights and accessories come on at full strength, but the engine will not crank or turn over unless I shift out of and back into Park.

Any ideas?

 

Codes? Sounds transmission range switch or shifter cable is out of adjustment. I would check shifter cable first. If sugary drinks are spilled on / in to console gummy shifter mechanism could be cause.