How long are you planning on going for?
How many people are going?
Do you have kids? What ages?
If you are only going to be going for a few days, my advice would be to stay closer to the city (Arlington, Alexandria, Falls Church area)
If you want to go for about a week, it might not hurt you to stay farther west in virginia (Herndon, Reston area).
The majority of the hotels in the herndon, reston and fairfax areas have free parking.
We lived in Herndon, VA for 20 years and by metro it would take us about 20 minutes to get into the city. It was convenient enough that we took the metro to George Washington Hospital for a year of medical treatment for me instead of driving in. It is also very convenient because to get from herndon to the smithsonian museums and the mall, you dont have to switch metro lines.
Some hotels I found in for the summer under $200 are:
Hyatt Fairlakes (Fairfax, VA)- about $115/night (free parking)
Hyatt Regency (Reston, VA)- about $200/night (free parking)
Residence Inn Fairlakes (Fairfax, VA)- about $180/night (free parking and breakfast)
Fairfax Marriott at Fairoaks (fairfax, VA)- about $180/night (free parking)
Hyatt Arlington (Arlington, VA)- about $175/night ($17/day parking)
Courtyard Alexandria Pentagon South (Alexandria, VA)- about $180/night ($10/day parking)
Most (if not all) of the hotels IN DC will charge for parking. A lot of the hotels in Alexandria and Arlington do as well. The best bet for lower prices and free parking will be in teh herndon, reston, fairfax area and in all truth, the metro system is very easy to navigate and it does not take very long (unless you hit rush hour (between 6 and 9 AM (going into the city) and 5-7PM (leaving the city) are the worst times))
I would advise that unless you will be there for 5 or more days, dont plan your trip around sunday or monday. Monday is when most of the smithsonian museums are closed.
If you have any questions about where anything is, best places to vists, etc feel free to PM me.