92 olds 88 royale, 3800, wont start after hot, dead key no sound, sometimes starts with a jump, security light not blinking.

Lots of possibilities.
If the alternator was not putting out 14 volts or more when reved up, then the battery could be draining as you drive.
But then you would still get a click from the soleniod, and jumping would work.

My best guess would be dirty battery terminals, since they can rapidly change with chemical reactions. They should be taken off, brushed shiny and greased.

But starters can fail from heat.
Another possibility is a safety switch could be bad, like neutral safety or clutch safety switch. Try neutral instead of park.
And the anti-theft system is also possible.