Generally speaking, no.
California Vehicle Code §22658 provides when an HOA may tow vehicles for parking violations. Specifically, only if (1) appropriate signage has been posted; (2) the vehicle has been issued a notice of parking violation and 96 hours have elapsed since the issuance of that notice; or (3) the vehicle is inoperable and the local traffic enforcement agency has been notified at least 24 hours prior to towing; or (4) the property upon which the vehicle is parked is improved with a single-family dwelling.
If you are a tenant of a dwelling within an HOA, then your rights are dictated by the HOA's CC&Rs and Rules and Regulations.
You likely have a valid breach of contract claim against the landlord if the landlord's failure to pay HOA dues is causing you to lose the benefit of what you contracted for in your lease or rental agreement.
Frank W. Chen has been licensed to practice law in California since 1988. The information presented here is general in nature and is not intended, nor should be construed, as legal advice for a particular case. This posting does not create any attorney-client relationship with the author. For specific advice about your particular situation, consult with your own attorney.
The short answer is no, I do not believe the HOA can do this. However, you'll want to check the HOA bylaws to ensure there is not a provision that would allow the HOA to seize a tenant's property. I cannot imagine such a provision exists but again you will need to check. You should talk this over with the HOA board and your landlord to ensure your car is not towed. Just because the HOA may not be permitted, does not mean they won't.
Legal disclaimer: This reply is provided for information purposes only and does not represent legal advice or an attorney-client relationship.