Hello SavorySkeptic. I completely understand your pain regarding the price of authentic Parmigiano-Reggiano, as it has indeed become quite expensive lately. To get that same sharp bite and umami flavor without breaking the bank, I recommend looking into a few fantastic alternatives.
First, Grana Padano is a very close relative to Parmesan. It is made in a similar traditional style in Italy, but it usually costs significantly less and has a slightly milder, yet still delightfully savory flavor profile that works wonderfully on pasta and in salads.
Second, Pecorino Romano is another brilliant choice. Made from sheep's milk rather than cow's milk, it actually packs an even sharper and saltier punch than Parmesan. Because of its intense flavor, a little bit goes a very long way, which means your block will last longer and save you money.
Finally, domestic or local hard cheeses often labeled as American Parmesan or a generic hard grating cheese can serve decently well in everyday cooking, especially when melted into a hot sauce or tossed into a heavy salad.