Those clever, helpful folks at Washington State Parks.
They are launching a series of blog posts with suggestions on how to get to various parks using public transit.
As they point out: "If you don’t feel like driving or simply don’t drive, you still have options to get out and enjoy your Washington state parks. Take a lunch, a friend, the family or even a bike. Many public transportation authorities have stops at or within walking distance of a beautiful state park destination. And when you visit a Washington state park by public transportation, you don't need a Discover Pass!"
For instance, they point out that King County Metro Transit can get a person (or family) to Bridle Trails State Park, Lake Sammamish State Park or Saint Edward State Park, and they link to Metro's trip planner to find your best route to these parks.
Their inaugural post on the subject focused on the northwest part of the state, from Seattle to the Canadian border. Here's the full blog post.