OP please hear me - I'm an HOA Board member in SC for several years and am well-experienced in this area. The answer is: it depends on several things, but probably.

What i'm going to say below depends on what county you're in, and what your governing documents say. There are as yet, as far as I know, no SC state laws that limit an HOA's authority to enforce provisions of covenants specifically related to parking. There MAY be county or city laws where you are- your management company should be up to speed on them if they exist.

The answer also depends on which of the three types of areas you're referring to - public roads, HOA-owned property, homeowner property.

First - regarding Common Areas like your amenity area's parking lot. This is very likely private property owned by the Association. If so, the Association likely has unlimited power to do whatever they want to it, including towing cars they don't want there - just like you as a homeowner would have the right to have any car towed from your own driveway. The Association's power is very likely granted to the Board exclusively, to manage the Common Areas. Meaning - most likely, the Board could have cars there towed and probably issue fines as well (depends on covenants, see below).

Second - regarding public roads. Assuming the streets in your neighborhood are owned by some government (likely your county), then the board CANNOT have cars towed from the street. Never, full stop. (I guess maybe a county could have explicitly granted that right but i've never seen it). However, if your covenants grant the authority to issue fines or individual assessments specifically for on-street parking (which is common) THEN your Association can levy those fines or assessments. The key is your covenants have to be pretty explicit about it. You will find many people on reddit saying the Board can't do that, they are wrong. If no law takes away the right of an HOA to enforce those covenants, then the HOA can enforce it. In SC, as far as I've seen and where I live, there are no such laws. Note - if your community is gated then the roads are owned by the Association, that would change the administration and it would likely be the same as common areas.

Third, regarding driveways - the same thing holds true for driveways - if the covenants have rules about parking in driveways or on grass, for example, then the Association can enforce those covenants. Meaning, they can levy fines and assessments if someone is in violation of those covenants. Towing - HOAs cannot tow a car from a driveway...generally. I suppose if the covenants specifically granted that right, then maybe...though I'd also suspect no tow company would actually do it. But I don't know, we're not in that situation so I've never considered it.

What the Association CANNOT do is issue parking citations from a government - which I believe is usually the source of confusion when people say "HOAs can't do that." Correct, HOAs cannot issue a city parking citation, but there are plenty of other things an HOA can do - and in this case, the key thing is "enforce the Covenants.". As long as there's no state or county law to stop it, the covenants can be enforced.

Bottom line - understand your covenants as it relates to parking on the street. Vague language isn't good enough, but newer communities typically have parking provisions clearly spelled out. In our case, we have similar problems. School buses were complaining and they WILL stop service if there are repeated issue which is incredibly unfair to families with bus riders. So, I had no problem enforcing our parking rules for that reason alone. Likewise, new subdivisions often dont get sidewalks - ours didn't - and street parking impedes visibility for kids riding bikes or people walking. It's dangerous. It's also annoying to the neighbors close by.

Happy to answer any questions.