What defines a public street in California? It’s open to the public and publicly maintained. Private streets, on the other hand, are privately maintained (by a homeowners' association or otherwise) and may be gated or otherwise blocked from public use.
Legally, it’s much easier to enforce your parking rules on private streets than public. Fortunately for your HOA, some or all of the roads in your development are likely private, meaning they are owned and maintained by the association and not open for public use. Your HOA may regulate parking for members and non-members alike on these streets.