How to cook Parmesan Cheese into a broth without making globs?
That recipe relies on starch exuded by the pasta to thicken the broth enough to emulsify the cheese. If your brand of pasta throws off less starch, the broth won't be thick enough to keep the cheese from globbing up.

I suggest adding some corn starch slurry just before the cheese. Remove a quarter cup or so of the broth, allowing it to cool slightly; add some corn starch (say, 2 tbsp as a starting point? I'll leave that to the group) and mix, then pour the slurry back into the main soup and stir for a minute or so over low heat, then add the cheese.

EDIT: Joe's comment made me realize something else -- you're not using pre-grated Parmesan, are you? You shouldn't use that for cooking. (Or anything, really.)