I realized that I hadn't followed up on this. I stayed at the Towne Place Suites LAX (Hawthorne) at the end of January. When I checked in, I said to the clerk that I objected to their new policy of paid parking. He didn't ask me if I had a car, and didn't charge me for parking.
When I went out for dinner that evening I noticed that some, but not all of the cars in the lot had a parking permit on the dashboard. I asked the clerk (same one as checked me in) if my car would get towed away if I didn't have a parking permit and he assured me that it wouldn't, but he handed me a permit anyway, so I put it on my dashboard.
Still no charge for parking, so I guess it's either the "honor system" or else they accept the "I don't wanna pay" as sufficient.