What actual effect does a Parmesean rind added to soup have? The answers to this question state that adding a Parmesan rind to a cooking soup will add a cheesy flavor to the soup.  

How much flavor does a rind really add?  Does it flavor the soup to a degree that it will replace freshly grated Parmesan?

Does it thicken or cloud the soup?
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.