Well, I think I figured out where to start -- Park City is in Summit County, so it would be the Summit County Recorder's office. Here's the link:
http://property.summitcounty.org/eaglesoftware/eagleweb/docSearch.jsp
Good luck with your search. Not all jurisdictions make this info available publicly, online and free, but many do. I read somewhere that the Gosselins had set up a corporation, so they may have their property ownership under that name.
I've searched all I'm going to. I don't care enough, and from what I can tell from skimming the blogs, no one knows for sure whether or not he bought a condo or a timeshare week. it's not been confirmed one way or the other. it's another one of those things we'll never know for sure. LOL