I agree that this does seem excessive, and it's not up to your HOA, or appropriate for them to take on enforcement responsibilities for the laws requiring insurance and requiring citizenship for DLs. I suppose it's to help the HOA deal with any insurance claims that arise in the parking area they have control over, and this information is easier to get before there's a problem then after, but I don't think that justifies it.
As for what they can do, you should check your CC&Rs to see what they say. I'm guessing they don't address an operational detail as minute as this, so you may try to politely decine to provide this information that you consider excessive on privacy grounds, and assert that the plate number, year, make, model and color of your car is sufficient ID to get a parking permit.
PLEASE READ THIS BEFORE YOU COMMENT, EMAIL ME OR PHONE ME. I'm only licensed in CA. This answer doesn't make me your lawyer, and neither do follow-up comments and/or emails and/or phone calls, and you shouldn't expect me to respond to your further questions if you haven't hired me. We need an actual agreement confirmed in writing before any attorney-client relationship is formed. This answer doesn't constitute legal advice, and shouldn't be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue.
The extent of the information sought seems excessive. The HOA can ask for personal information, but the homeowners and their tenants and/or guests do not necessarily have to provide all of the information. There does not appear to be any California law on this issue as it relates to common interest developments (HOAs), such as the Davis-Stirling Common Interest Development Act.
The question will really come to light if and when the HOA board refuses to issue a parking permit because the applicant refused to provide personal information beyond what is reasonably necessary.