The plan is too ambitious even though you should have a great deal more time than you state.

Parking in DC is brutally expensive, when it is available at all. Most garages charge the full daily rate after an hour (anywhere from $25-50). As you move around, you will be paying that several times.

A better plan is to grab your luggage and then Uber to a hotel nearish to the first place you are headed. Drop your bags there (a very nice tip to the bellman will do it). Should run you about $15 into DC.

You have not told us what carrier you depart on, but I cannot imagine a reason to be at IAD 3 hours prior to departure unless you are planning dinner out there. Allow yourself 1:15 (0:45 + rush hour). Uber back out to IAD will run around $40 (Maybe $45 depending on where you are when you need to swing back to the hotel to pick up your bags).

You can use Uber, the Circulator bus ($1), or Metro ($2.25) during the day if walking isn't in the cards.

All of this will be more pleasant and cheaper than worrying about parking and waiting for your car to be brought around.