The photos on that web page showing Bird's three seemingly different metal mouthpieces are all photos of the same mouthpiece, the brass made-in-England Selmer, with different ligatures. Nicolas Trefeil has a good discussion of this at http://www.nicolastrefeil.com/cparkersetupandmpc.htm, including a mention of the original ligature's unreliable design. I have never seen a Runyon that looks like Bird's English Selmer, but maybe someone can post some documentation showing that his Selmer mouthpiece was actually manufactured by Runyon, since it's not clear who, in fact, made it.
That photo is a bit blurry, and maybe the shank appears somewhat rounded, but there are pictures of this mouthpiece-ligature combination in Chan's book To Bird, with Love that make it clearer that it's the English Selmer. The only photo I've seen of Bird with what looks like a Runyon is this one: http://library.umkc.edu/spec-col/local627/images/stomp/charlie-parker-crop.jpg.The mouthpiece on the bottom is definitely different. The shank is rounded. The only reason I am guessing that it is a Runyon is that I saw one just like it with the same ligature on eBay (didn't win it)