I don't know the national parks specifically, but just about every trailhead in the mid WA cascades, with the exception of a few that you need additional permits for (wonderland, enchantments, etc)
Most of those are on national forest land, which is different than a national park. The wonderland is at Mt. Rainier National park, which requires permits for all overnight backpacking. Same with Olympic and North Cascades National Park.
