The 3 weeks parking is available only with the weekend rate ($159/night). You get charged an extra $25/week for parking beyond the provided period. If you were going out of IAD, the Marriott by Dulles has a park and fly package that gives 2 weeks of free parking, irregardless of a weekday or weekend rate.

If you are flying out on a weekday and coming back on a weekend, you could ask the hotel if you could get the parking pass first and stay at the hotel after you get back. The SpringHill Suites BWI will allow you to do this if you make a nonrefundable deposit for the room. This would allow you to get the weekend rate.