On a recent trip I parked in the town of Bishop, in the public lot behind the police station. Walk into the police station and at the counter they will take your car and trip info, and keep track of your car. Although it's not in the police station lot, it's just a few hundred feet away and not likely to be vandalized. In addition, there's something very satisfying about a police station with such customer service!
About starting at North Lake vs South Lake -- I agree that Muir Pass and the lakes just north-west of it is stellar, worth repeating again even if you've crossed Muir Pass five times already. If you've got easy class-2 hiking skills, you can go in via Lamarck Col from North Lake (as described in Secor). Saves a lot of miles compared to entering via Piute Pass.