First of all, I don't know what those folks at Formaticum are smoking, but Parmigiano-Reggiano (PR) should NEVER be stored in paper. Generally speaking, PR is at it's best when you first buy it. It's a little bit creamy, bursting with flavor and relatively easy to grate. Some might feel differently, but, for me, that's the peak. It really depends on how long it's been in the case for, but I think you can store it for a week in just about anything and it'll still be top notch. Beyond that, plastic wrap should buy you a second week. Plastic is air permeable and air is the enemy of PR- air dries it out and it loses much of it's flavor and becomes almost impossible to grate. The thicker the plastic, the less air permeability you get, so shrink wrap should buy you more time, but I don't think it will buy you more than a month.
I use PR in lasagna and meatballs, which I don't make often- maybe once every three months. My source for PR doesn't always have it available, and, when they do, the chunks often have rind on two sides. Because of this, when I see a chunk with rind on only one side, I buy it, even if I'm two months away from baking with it. I don't like spending 12.99/lb. on PR and having it dry out as hard as a rock, so I've come up with a method that gives me two month old PR that tastes as good and grates as easily as the day I buy it.
Glass. It's really difficult getting a hunk of PR into a glass jar, but the glass blocks out air for a very long time. I use Pecorino Romano at an even slower rate- I grate about two tablespoons a year, so I store that in glass as well. I just checked one of my jars in the fridge a few minutes ago that's been in there at least 3 months- flawless. Glass jar, twist metal lid with a rubberized seal. Old PB jars seem to do the trick.
PR stored in glass will tend to grow mold, though. One thing to be aware of is that PR, as it's produced, grows mold- mold that the cheesemaker wipes off with a damp cloth. I feel a little better scraping a fraction of an inch than I do wiping, but I still feel that the mold is fairly benign. I have noticed, though, that if you really let the mold go crazy, it will penetrate into the cheese, and you'll find yourself hacking large chunks off of it. I've taken steps to avoid mold, and, although the results are still preliminary, salting the cheese before it goes into the container looks promising. Just a light sprinkle. It doesn't seem like the cheese absorbs the salt- and it can wiped off prior to grating.
I think if I really wanted to guarantee the longest possible life for my PR, I might get my hands on food grade hydrogen peroxide and sterilize both the cheese and the jar- like they do for juice boxes. But for now, a sprinkling of salt seems to buy me some time.