Yes. A well aged Parmagianno Reggianno WOULD have umami. Not sure about the other, softer cheeses, though. It's all about the amino acids and glutamates. Parm often has little white crystals of tyrosine, which are pure umami! Like MSG. Umami is more often than not a savory taste. There's a great book about it called "The Fifth Taste
" which goes into amazing detail about it. Here's a review/overview
from the Art of Eating that goes into some detail. It will tell you much of what you want to know about umami.