How long does parmesan sprinkle cheese last in the refrigerator?
Use-by dates are set as the very minimum time that the product in question will be good for, provided that it is stored properly.

In addition, and as a general rule, because Parmesan is a dry, hard cheese, with very low water content, the use-by date is almost always far shorter than the actual time that the cheese will be good for, again, provided proper storage.

By "parmesan sprinkle cheese", I assume you are referring to the ones that come in a parmesan-wheel shaped box, such as this:



Their shelf life is far longer than their expiry date, and I would not be worried about using it beyond that date. However, for those boxes, most manufacturers tend to use the youngest cheese possible (for Parmesan this means 12 months), meaning it has not had as much time to age (some parmesan is aged for as long as 36 months, and some manufacturers age it for even longer than that), meaning that there isn't much flavor in it.

Personally, I stopped using the boxes a few years back, when I realized that I used far more than I would if I simply used a grater and a piece of parmesan, meaning that it I spend less (if memory serves, I calculated the annual savings to something like 300 Norwegian Kroner/50 US Dollar) on parmesan, getting more, and I think better, flavor from it.

It should be noted that my family uses a LOT of parmesan (depending on how often we have guests, we go through anywhere from 1/4 kg to 1 kg per month), so the savings argument may not hold true for you.