I found the following info on official page of the Museum of the Confederacy:
"Parking is available in the MCVH Visitor Patient Parking deck at the end of Clay Street. Bring your parking ticket to the front desk and we will gladly validate (...)."
I parked my car there, just by the tall building of MCVH which meant the Medical College of Virginia Hospitals (on my picture) and I validated my ticket at the desk all right. Despite it when I was leaving the parking I had to pay $4.00 to the guy at the parking gate. It seemed that he didn't write the webpage of the museum.
With all the construction going on downtown, it is a challenge to legally park at the weekend. We used to be able to park along Governor's Street near Capitol Square, a central location between the Museum of the Confederacy and Shockoe Slip and Bottom, but construction there temporarily closed that. Trying to find a cheap (not metered) legal parking slot during the week is akin to finding icebergs in Rio. Christian and I came to the conclusion that the city's favourite sign was "No Parking."