Hi Kate, First comment is you are not alone! This is a prevalent problem throughout the US. I am in Arizona and it's close to the #1 complaint in the online reviews (scorpions may edge it out a smidge!).
Next, many of the comments here are correct- if reviews haven't started on the issue, they will. It's good you are working on getting ahead of it so when prospects inevitably ask about what they read, you can confidently and correctly say you've implemented X,Y, & Z to address the issues.
So now on to some ideas. 5-6 adults, each with vehicles, in one unit is untenable. Beginning now for new rentals and upon renewal for existing residents with sufficient notice, I would suggest you implement a maximum # of vehicles or maximum # of parking passes as others have suggested. Parking passes are a must, at the least, if you don't want to go fully down the 1 assigned spot per unit road. You need to know which vehicles are resident vehicles (versus guest or unauthorized occupants). I would hold off on the towing agreements for a bit as those tend to open up a new can of worms you don't need to worry about right now. Also, I agree with others that garages must be used to park a vehicle (perhaps you can ease some of the friction from this by working out a cooperative marketing arrangement with the closest storage facility).
I haven't seen any comments on this yet- Have you measured to see if re-striping is an option? Before you remove HC spots, I would have someone measure for re-striping the others. I've seen sites gain as many as 10 spots doing this. Now, it's possible you start to hear about car dings with this plan because the spots are smallllll.
Good luck! I think it's great you are getting a plan of action together now. Let us know how it goes!