What actual effect does a Parmesean rind added to soup have?
What parmesan rind adds isn't so much cheesiness (an extremely difficult to pin down set of flavors, considering how many cheeses there are), but rather "umami".  Umami is thought of as the "fifth flavor", after sweet, sour, salty and bitter.  It's normally described as savory/musty.  Common foods with umami flavors (other than parmesan rind) include mushrooms, beef, anchovy, seaweed, soy and fish sauces, and (somewhat less so) tomato pastes.

In the right recipe (beef stews, coq au vin), a blast of umami can be amazing.  Parmesan rind is subtler than that, but still a good tool to have at your disposal.