[Mike]

The wall to the left is grade 1 listed, so that probably can't happen - the actual green isn't protected except the statue on it, but the impact on the visibility of a G1/2 listed structure would probably lead to a planning refusal.

historicengland.org.uk 

At the time the car park was constructed, while it was listed the protection for listed buildings and structures wasn't as rigidly enforced as it is now.

Personally I think the current solution is ideal; but I also feel the needless OBs from parliament are a waste of resource. They bring nothing much to the story; most prospective MPs are in their constituency during the GE, arguably Downing Street and Buck House are more important places to show any action.