The HOA is constrained by the Declaration/Restrictions. If the Declaration requires cars be parked in the garage then you are going to have to comply with that. I would suggest that the language you provided is pretty clear as to their intent.
if your HOA declarations allow it yes, HOA restrictions may impact common property but almost always are applied to the private property of owners under the HOA.
This answer does not establish an attorney client relationship, nor does it constitute specific legal advice.
Even before the association documents, make sure to check your survey to be sure that the driveway area in question is included within your property lines.
Please note that the above is not intended as legal advice, it is for educational purposes only. No attorney-client relationship is created or is intended to be created hereby. You should contact a local attorney to discuss and to obtain legal advice.