Hi Lisa, I am pretty sure you can do this easily in K2, I would try creating A Category Parks then a Category Regions as a Sub-Category of Parks then creat the rest of the cats (Intermountain, Pacific West etc.) as Sub-Categories of Parks. Then create items for each category.
To display the same as your link Create Joomla Menu Item (K2 Category view) for the Parks Category.
And in K2 adjust Category item layout to suit making sure that in Other layout options - Catalogue mode is set to no. Also in Sub-category handling make sure Sub-category title is set to show. (For some reason these titles are not displaying on my development site, I just made a post about this issue, but I figure this setting wouldn't exist if it didn't work.)
Hope this helps Pete.