When I first start the car after it has been setting over night, the starter grinds; I turn the ignition off, try again and it starts OK. It starts properly once the engine has warmed up. I've replaced the starter and cleaned the battery connections. I've also tried starting in neutral, and it still does it. It first started when the temperature dropped below 32F. Now it's Spring and does it regularly. Is this a weak solenoid or maybe a bad connection somewhere?
PS: No April Fools. I'm serious.

Sure sounds like a weak solenoid.
If it was missing teeth on the Flywheel It wouldn't get better when warmed up.

If the flywheel is missing teeth, it'll screech like screaming eagle.

I would like to thank yodawill200 ane driverdriver for helping me solve this.
I checked the wiring and connections as well as the pinion gear on the starter and the few teeth I could see on the flywheel: everything seemed alright.
Lastly, I checked the resistance across the 2 pin connecter on the inhibitor switch. In park and neutral the switch measured up around 400 ohms.
I loosened the 3 adjusting screws and turned the switch as far clockwise as I could. Then I measured 2 to 50 ohms on park and 2 to 10 ohms on neutral.
I observed the ohms as I played with the shifter in the car to find the low ohms.
I need to either replace or repair the inhibitor switch, however, I haven't had any grinding from the starter.
P.S. The inhibitor switch is spendy!

I'm having the same problem how exactly did you fix this?

I'm confused on how the inhibitor switch would have anything to do with the grinding