Parking is fine at all times. It never seems to be too busy. Truly an underappreciated park. Be aware that when driving, check for your arrival time since some entrances are only open during some hours.
There's LOTS of parking all over the arboretum. There's a lot by the visitors center, a huge lot just inside the NY Ave entrance, and other smaller parking areas by all of the exhibits. You can drive your car all through the arboretum too!
Plenty if parking all free.
There is parking. I went in mid march. There weren't many visitors then.
Thanks! Your answer is awaiting moderation.