To keep parmigiano-reggiano, or Parmesan, fresh for as long as possible, wrap it in wax paper and seal it in foil. Make sure that it is separated from other foods so that it doesn't absorb their odors, according to FineDiningLovers.com.
It's possible to wrap the cheese in parchment paper or plastic, but wax paper is preferable because it preserves the proper humidity for the cheese and keeps it from drying out. After wrapping it in wax paper, cover it entirely with foil, which protects it while still allowing some air to circulate. It's best to store Parmesan in the bottom of the fridge at a temperature between 39 and 46 degrees Fahrenheit. Never freeze Parmesan unless it has been already shredded.
If your cheese has been exposed to the air too long, Tammilee Tips gives a useful way to revive it. Soak a cheesecloth in water, and wrap it around the cheese. Then, wrap it in plastic wrap, and keep it in the fridge overnight. In the morning, the cheese will be fresh again and ready for your next recipe.