You can park at limited metered spaces near the building, but down the hill from the State Capital is free parking. This parking area though is normally used by State employees during working hours, so plan your visit to the War Memorial around this. There is free parking at the Bicentennial Mall as well.
Parking is limited. Take a lyft if you can.
There are meters close
There is street parking nearby
After 4:30PM on weekdays, you can park in the State numbered lot straight across Charlotte Ave from the Tennessee Tower. There are plenty of spots. It is a block and a half walk from there. Otherwise, there is no parking close by.
I find it easiest to take a trolley or taxi to the state museum and then just check out the memorial after the museum. The museum is right across the street.
Depending in your timing it could be pretty easy. We just happened to visit on a Sunday morning on the way out from vacation in town. Easy parking everywhere
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