It is actually very rare to find a ski in/out condo or home that rents long term. Because those properties are the most sought after by vacationers, most owners only rent nightly. Though Park City is very pet friendly, there aren't that many condos that allow pets. The ones that do mostly only do so for owners.
Racquet Club is one of the few that allow pets for owners and tenants and they often rent under 2K per month as well. They are in Park Meadows, so not ski in/out, but reasonably close to Park City resort.
Your best bet is to check http://www.ksl.com
The Park Record or craigslist for owners who are renting long term. Another possiblity is to check VRBO. You might be able to find an owner who is willing to rent longer term that nightly on that site.
Equity Real Estate Luxury Group