Not knowing specifics about the community, I'd have to say the HOA can shove off! I work in land development and there are basically 2 types of land - public and private (duh!). Most streets are public with the small exception of private streets, which could very well be the case here. I've designed a few apartment/townhome projects and most are private roadways, of which usually the HOA is responsible for the upkeep. Not sure if that puts them in the position to prevent OC, but I know my HOA doesn't allow any parked vehicles in the streets that can't be moved upon request.
I'm not sure of the extent of the powers of a HOA, but I've seen some that pull enough weight to basically govern a small town. (cough, cough....Brandermill....cough, cough).