Looking at the wiring schematic in a little more detail the relay is the most likely culprit. It would supply the park voltage after the intermittent signal starts the motor rotation. The motor and relay positions determine the activation of the park circuit. The symptoms you are experiencing are indicators that the park cycle in the intermittent mode is not being activated. I still would not rule out 100% on the multi-function switch or CTM.
I don't know what series level your truck, standard, SLT, or SLT+, but on my 02 SLT+ there are three different CTMs, depending on the option configuration. They are no longer available from commercial sources (dealer, parts stores, etc). I have had alarm issues with mine and have to live with it, or find a source and take a wild chance that I will not inject a multitude of other issues since it controls lots of stuff.