Your question raises many troubling issues for a literal pilot besides the straightforward weigh/volume one. Are you intending to grate your own block of Parmigiano-Reggiano cheese or purchase a canister of pregrated pretend-Parmesan cheese? (Can you tell from our tone which we recommend?) And if, as nature intended, you do grate your own cheese, will you pack it into the measuring cup or gently spoon it in? The difference by weight will be substantial.

In this country - as explained elsewhere in an answer that the questioner found to be grotesquely offensive when we meant no harm - virtually all cookbooks and recipes measure ingredients by volume rather than weight. The exceptions are cookbooks for professional chefs and bakers, which recognize that measurement-by-weight is a much more consistent measure.

And no matter how much of a literalist you are, it's time to recognize that you're adding Parmesan cheese to a dip, not landing a 400,000-pound airplane, so add as much or little cheese as you like, until it tastes great!