I've had the brushes wear down inside my starter, turn the key to start and nothing. I've also had one brush not make contact and the starter sucks down the amps but doesn't move (lights dim). It made a sound like frying bacon, but never spun up the starter.
It may be something else, but starter brushes are my go to when the starter refuses to work, then the neutral safety switch.
Tapping in the side of the started with a hammer, will often get you a few more starts on worn our brushes. Trying the shifter in neutral instead of park and maybe jiggling the shifter may give you some contact through the neutral safety switch.
My last go around was actually the battery, one cell internally was acting up, it had about the same voltage as yours, but wouldn't pass enough amperage to get the job done. It also messed with my IAC, the IAC got flaky because of the bad battery, the motor would stall when I let my foot of the gas pedal sporadically and sometimes wouldn't start (with the starter turning over) without giving a little gas pedal to get some air to the motor. Since the new battery, both the starter and IAC are doing fine. If it starts with a jump, it may be a bad or poor cell in your battery.
I've had the oil leak from my oil filter adapter "O" rings onto my starter and find it's way inside. Enough to cover the brushes in oil and cause iffy satrts.
I've also driven through muddy water and had it coat the brushes, they dry and the muddy water leaves a crust on the brushes. Sometimes just cleaning the brushes is enough.
I've replaced a couple of starter relays, one was full of mud, the other full of rust.
I've never replaced a starter, I've still got my original from 1987, but I have swapped out more than a few sets of brushes.
Just some ideas.
