You didn't come out and say it, but I'm going to ask, is this a "commercial vehicle"? That may be a sticking point. Generally, a "commercial vehicle" is one which is either registered as such, or one that displays some sort of advertising/company identification, etc.
The way you posted your question in the first place (not mentioning the type of vehicle), makes me think that you probably asked that exact same question to the code enforcement folks and did't mention the type of vehicle it was.
For now, assume everything else you said is accurate, particularly that the HOA doesn't own the roads.
Around me, parking of commercial vehicles in residential areas is regulated by *zoning*, not code enforcement. Overnight parking of a commercial vehicle in a residential zone is NOT permitted, and can result in fines.
One solution to that would be to park that vehicle inside your garage, if you have one.
You might want to check local ordinances about that.
That being said, if, in order to get to your house, you need to cross over any HOA-owned property, they would likely have the right to deny your commercial vehicle access to that property.
I'll bet ya a nickel that for some reason you've managed to tick-off a neighbor, either in regards to the truck parking, or some other issue.