Can I continue to age store bought cheese?
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.