I am not aware of any "paid" parking lots in the area near the Greenbelt metro station. There was an article in today's Wahsington Post (WP) that the parking lot at the Greenbelt Metro is going to be for buses ONLY on Inauguration. There are a number of churches in the Greenbelt area. Perhaps you could contact them and make a "deal" for a small donation.
FYI the WP has a website dedicated to Inauguration
www.washingtonpost.com/inaugural Personally, I think that any traffic plans announced at this point are going to be subject to change. I don't think the traffic plans will be finalized until after the first of the year.