There are times when you're not focusing on anything specific, hence why you get ideas / breakthroughs in the shower, or when looking out of the window of a train / car or just mindlessly walking, because you're not focusing on anything and your System1 can solve stuff.

Though I feel that System1 can't work while System2 is focusing on other stuff like playing games or browsing social media so I don't think they're necessarily running in parallel.