I know that many cheeses are considered to be better when aged. After I've purchased a nice block of Parmesan or such, can I continue to age it at home?
Is it practical? Will I get the same result?
Continuing to age a "nice" block of Parmesan is not going to do anything for you. It has already aged for over year and has changed pretty much all it's going to.
Similarly aging cheap, canned, Parmesan-like product that is aged only a month to cut costs will also not be good because it has too much surface area and will oxidize. It isn't very good to start with of course.
Where this will work very well is with cheap, young cheese. Buy an inexpensive block of young cheddar that doesn't have any crazy additives. Cover it and let it sit in the fridge for a month (or more). The bacteria in the cheese will continue to munch on available lactose and turn it into lactic acid. If you ever see mold wipe it off with a paper towel and a little vinegar.
At the end of that time you will have a much sharper and more valuable block of cheese. You've traded your time for money which is the trade-off cheese manufacturers always make.
The answer here is yes and no.
Yes, aging cheese at home will often improve its flavor and texture. No, aging at home is not going to turn a block of Joseph Brothers Cheddar into Cheshire or (as the other answerer points out) a wedge of Stella "Parmesan" into Reggiano.
First, you want to deal with a moderate quality cheese which is simply not very aged. Really cheap cheeses tend to be full of added vegetable oil and "processed cheese food" which is incapable of ageing beneficially. So, for example, Tilamook or Spring Hill cheddar would be good candidates whereas Safeway Select would probably not be. Likewise, Rumiano or Argentine Parmesan are better choices than Kraft.
Second, you need to pick a cheese which ages well. In general, this means a harder cheese; from the harder side of semi-soft (cheddar) to hard-as-wood (parmesan). Cheeses which are not usually aged (brie, feta, chevre, etc.) will just become moldy and slimy.
Third, you need to deal with rind. Ideally, you'd use an "intact" cheese with its full rind or other covering to age. However, most such cheeses are ones which don't age well, such as brie and crottins. There are, however, a few wax-covered cheddars and goudas which are less than 4lbs in size and sold whole (make sure it's actual wax and not plastic). If you have to buy a piece of cheese rather than a whole cheese, then you should get yourself some cheese paper, a semiporous paper which keeps the right balance of moisture and airation in most cheeses. Take the cheese out of its plastic and wrap it tightly with a single or double layer of cheese paper.
You could potentially experiment with covering the cheese in wax yourself, but I don't have any information on this.
Fourth and most difficult is refrigeration. Most cheeses like to age at temperatures between 50F and 60F, not at the standard US fridge temperature of 40F. This means having a separate fridge for your aging cheese, something which takes this whole idea out of the realms of economy. Some folks have reported moderate success with aging at 40F, but be prepared to have no detectable difference in your cheese after several weeks. That's why we keep the fridge at that temperature after all! Needless to say, aging your cheese at room temperature (or worse at unstable, fluctuating temperatures) is not recommended.
Also, a standard fridge is far too dry for aging most cheeses. This, however, is easily solved; simply get a very large plastic food container (at least 4X the volume of the cheese(s) you will put inside), line the bottom with a bamboo placemat or similar aerated/draining surface, put the cheese inside and put the lid on. This will also keep your whole fridge from smelling like old cheese. If you live somewhere dry (humidity < 50%) then you may want to put a wet damp paper towel underneath the bamboo to supply moisture. Depending on the length of time you're ageing, you might want to open this box once a week or so to change the air in it.
Most of the above advice comes from Culture Cheese Magazine, which I strongly recommend to anyone interested in cheese and cheesemaking.
I have been aging store-bought mild cheddar cheese at home now for about 25 years. I just purchase a two pound block of Tillamook or other mild to medium cheddar, leave it completely sealed in it's wax or shrink-wrap plastic, wrap it in a couple of layers of aluminum foil, the a couple of layers of newspaper. I tape it up securely and place it in a low cupboard, out of the kitchen area (so that it's in a slightly cooler location). Anywhere from 3 to 9 months later, I retrieve it, unwrap, drain the liquid off, and enjoy. It's deliciously sharp with no mold whatsoever.