As a landlord, I wrote a post a while back based upon how I post properties for rent. I choose the free services like craigslist since I manage them myself. Recently however, I saw a For Rent sign in my neighborhood and couldn't find it listed on the free sites, so I did a bit more research . . .
Some owners hire property management companies that do not always list rentails on the free sites. These are a couple prominent long-term rental property management companies here in Park City:
Angel Properties: http://www.angelpropertiesparkcity.com/Angel_Properties.html
Coalition Management: http://www.coalitionmgmt.com
If you're moving here to work in Park City and you just can't find anything in your price range, other neighboring communities like Heber City, Midway, Kamas, and Wanship are about about a 20-30 minute drive away via major well-maintained roads, and Salt Lake City is about 30-40 minutes away. Each has more affordable options than Park City, though there is little or no bus service from there to here and so you'll need either a car or a steady ride.