Not sure if it is the same for your year model, but in the 2010 Silverado the power to the parking light fuses is fed by the park lamp relay that is operated by the BCM when the light switch is operated.
The relay is located in the under hood fuse block.
Here’s an image taken from the 2010 Silverado owner manual showing the location of the relay in the fuse block and also an image from the 2010 Silverado electrical wiring circuits showing how the BCM (blue box) operates the relay (red box) to supply the fuse(s) (green box).
(click on image to enlarge for better viewing)
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