I guess my 2004 Silverado isn't new enough for the ignition switch to trigger all the lights. Only the front DRLs come on, and then only if the engine is running and the transmission is in gear, unless it is dark, then the head and taillights come on and the DRLs turn off.

The easiest way to get the parking lights to come on at both park and headlight positions is to just move the wire from the parking light position on the light switch to the taillight position, as was mentioned earlier.