When you click on links to various merchants on this site and make a purchase, this can result in this site earning a commission. Affiliate programs and affiliations include, but are not limited to, the eBay Partner Network.
Weird stuff like that is generally a ground loop. In this case it sounds like the ground connections for your dash stuff are more resistive than the ground connections for the brake lights. So when you complete the "circuit" by applying the brakes, the ground connection through the brake pedal switch and the light bulb of the brake indicator is at least more conductive than the one for the dash stuff, and the dash stuff wakes up a bit.
TL;DR, check your ground connections everywhere you can and clean them up.