A bit late to the party, but there is a way to change the fill color for this PNG in Illustrator. I use CC, so the controls may be a bit different from what you see, but in case it helps, here is what I would use.

 1. "Object" » "Image Trace" » "Make"
 2. If needed, go to the "Image Trace Panel" to adjust things until you have what you need. The first thing I'd recommend is checking the "Preview" box, then selecting a "View" option that shows outlines. For this specific file, picking "Ignore White" may also help. There should be functional equivalents for these in CS6.
 3. When the outlines are in the right place, close the panel then go "Object" » "Image Trace" » "Expand"
 4. Check your Layers panel. You should now have a Group with your new outlines. Go into the Group and select the items whose color you'd like to change.
 5. Your normal controls for changing fill and outline color should show up (if not, make sure you are using the Selection Tool).

As other folks have said, Photoshop is definitely easier, but I'll just leave this here in case anyone else wants an Illustrator option~