My hometown, Indianapolis, has privately run parks. Once upon a time, the city used to run golf courses as part of its park system, but about 15 years ago, it contracted out the management of the golf courses to private companies. It's worked pretty well, at least from this user's perspective (the courses have been much better maintained since they went private). Here's a brief article discussing the contract, which was recently up for renewal.
http://www.ibj.com/city-ready-to-award-new-golf-contracts/PARAMS/article/15212
Contracting out the operation of certain services nevertheless implies government ownership....