Won't start starter is turning over but not connecting with flywheel..?

Doesn't matter. The forks inside the starter may be missing or broke off or bad spring in starter that pushes the gear out. Try new started. Starter may have to be shimmed to make contact with flywheel. Starter may be kicking out but not engaging flywheel.

If the starter motor is turning the problem may be a broken flywheel tooth. If starter is not engaging the solenoid may be bad or not hooked up but that seems unlikely since starter was replaced. The problem's down there. If solenoid not clicking then problem may be further back.

If you mean the starter grinds from time to time and won't crank the engine when it happens, the problem could be a bad spot on the flywheel and/or a faulty starter. The starter uses a magnet and bendix gear to push the starter gear into the flywheel. If the magnet or bendix is weak, the gear will not engage and can damage the teeth on the flywheel.
You need to have the starter removed and check the flywheel teeth for damage.

I can't imagine how the drifting in park and the starter won't crank the engine could be related.
It is possible something happened to the flywheel and it is not now connected to the crankshaft, or the bendix gear on the starter is not engaging the flywheel.
But the transmission has a pawl or gear that locks the output shaft of the transmission when the shifter is in park. That gear could be damaged and would allow the truck to roll in park.
And the parking gear would not have anything to do with the starter issue.
I think I would check to see if the flywheel is connected to the crankshaft first.
Is the truck still under warranty ??

If the starter is spinning when you try to start but doesn't spin the engine over then you most likely have teeth missing off the flywheel ring gear. The facts that you say you have to push it and then it starts means the flywheel position changes and the starter has good teeth on the flywheel to engage. To confirm this next time the engine won't crank over remove the starter and look inside at the flywheel for missing flywheel teeth.

Your battery is to low on power it needs a good charge, or replace it is the best option for a brand new one!

It's your starter. Find location of your starter. By the flywheel. Then. Bang on it . Your car will start. You still need to replace the starter.

Sounds like you are not getting enough current from the battery to the starter. If your battery is fully charged, make sure the the battery cables are tight and corrosion free (baking soda & vinegar will clean the terminals, rinsed with clear water afterwards) and make surw connections on the starter are tight. There is also a starter relay in the circuit that could be bad. You could use booster cables to connect battery direct to the starter to see if it turns the engine over, but make sure polarity is correct. Put one end of the booster cable on the the plus side of the battery, then take the other end and touch the plus (+) side of the starter. If engine turns over (it will not start) then problem is either cables, connections, or starter relay. If it doesn't turn over, I suspect the battery itself or the starter.

Do you have the correct starter and is it grinding, this sounds like a bad flywheel or wrong starter or even a bad one. look at teeth of flywheel and see if they are damaged. good day

Did you also change the starter solenoid when you changed the starter? That is a possible cause.

Can you turn the engine over by hand from the crankshaft pulley to make sure the engine is not frozen or blocked? You will need a large socket to do that, but if you can't turn the engine over by hand, the starter won't be able to turn the flywheel either. The one click is probably the solenoid firing the starter piston to the flywheel but then nothing makes me think the engine is frozen in some way.

But if the starter and solenoid and battery is trying to turn a frozen flywheel, you should notice some drawdown of light intensity. Do you notice any change?