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

Now the kiddos will be real proud of daddy. 
I just read something too about his alleged reality show 'Divorced Dads' (insert retching sound here). Maybe we should start calling Jon Octodad. I read that somewhere that I can't recall now! 