This weekend I was finally able to make some headway on the project again and I was able to get my sample map rendered using Swing. Here are the pictures:
Map as rendered by TMX-S-Parser in Swing
Map as rendered in Tiled (Objects hidden)
As you can see, the basic layer rendering seems to be working now. I'm actually quite excited to start working on the other elements that can be in the map - objects, polygons and the like. I also have some ideas for 'properties'. If you'd like to see the code, check out the TMX-S-Parser project on Github.