If you can't legally drive to the spot, you can't legally park there.
This was my hunch and is one reason I thought I'd ask if anyone knows. To respond to CALSGR8, no, there are no signs at all, which is why I wondered about it. Had there been "No Parking" signs, I wouldn't have needed to ask. The one thing about what blue911 notes is that it's not readily apparent whether it's illegal to drive over the asphalt area behind the tree, but because of the metal barrier across the street my gut tells me that it is. That's one reason I didn't take the chance.
I've found a legal place to park for free, or for minimal metered rates, so I think I'll stick to that. Thanks to all who replied. I guess my general rule of thumb in the District—if you have any doubt, don't park there—is probably still the best way to go.