Dave from Kirkwood on 9/25/2018 8:14:40 PM:
Based on this website and personal memory, there are no bathrooms at the Research Park parking area. Heading west, there is one at the Weldon Spring trailhead (with parking). Next, Defiance has restrooms at the bike shop and restaurants (parking as well). The Matson trailhead has both restrooms and parking. The boat ramp area before you reach Klondike also has both. One final option (before Augusta) is to bike/walk up the hill (using paved path) to Klondike Park, which has restrooms next to the camping area.
Hope that helps.