1988 Cadillac STS Starter

The car will turn over after it sits for a while, but only for one or two tries. The starter is working fine since I took it off and tested it. I don't think it's the parking switch either. I've replaced the steering column and switch. The battery is very good.

Any ideas? I take pride in the fact that I can fix almost anything but now I'm running out of ideas. I imagine it could be some kind of module or maybe even the computer itself.

Thank you for any advice,

Ken Keeton.