99 Limited, 266k, Factory alarm? No CEL
Back history, Randomly the alarm would start, flashing headlights n Horne until the battery died. Once the battery was charged the truck would attempt to start with no key in the ignition. Unk if this fixed

Today’s new problem
In Park
The motor will turnover, start, the starter stays engaged

Shift the transmission into Neutral, the starter engages.

Shift the transmission into reverse or Drive, the starter disengages

If I switch the truck back to into neutral, the engine shuts off.

Once the The truck has been running for 15 minutes or more in drive, you can put it back in park or neutral and the starter will not engage. However the truck will shut off immediately.

Being a ltd, you would have Toyota's Immobilizer system, it should say so on the ignition ring. Has someone added a remote start and/or piggy backed an alarm system. If not you will need to go over your ignition wiring and look for a open short to open power.

Only thing on the ignition ring is Toyota Security. It also has the red LED security light on the upper center console.

There is no secondary remote or starter. I have looked everywhere

I would tend to agree with Malcolm99. Check for an aftermarket alarm or remote starter. Sometimes they will fail and the vehicle won't start or will start when not turned on, or the alarm will go off for no reason. Also check starter relay, park/neutral switch and ignition switch although they aren't likely the problem.

Did you pull the driver's side knee bolster panel down? Typically that lower removeable panel has to be removed to install or remove a remote starter.
I did have a Corolla do something very similar a while back and it turned out to be the park/neutral switch. I think the key had to be turned on though. It would start itself as soon as the key was turned on and the starter would keep running until put in reverse or drive, it would also stall when put back in park or neutral. It is the only vehicle I have ever ever worked on that acted like that due to the neutral safety switch.

Would it be possible to post a pic on here of your wiring?
Especially around the fuse box area?
Except for the neutral start switch, which only cuts power going to the starter circuit. It would take human intervention to bridge, and power that circuit. Toyota back in the day didn't have remote start as an factory option unless it was added on later as the above said.

Fuse panel

I can only get 1 pic to work

Attached Images

 

Does it do this with the valet key while the ‘smart’ keys all have their batteries removed?

I dont have a valet key. Only have 2 master keys

Did you try removing the batteries from both keys then?

The immobilizer keys do not use a battery. Nothing to unplug.

The images should be posted to an external hosting site and linked here if you want high resolution to work (imgur or similar).

-Charlie

Oops, my bad. Thanks.