How to make cauliflower pizza crust without using parmesan cheese?
As far as I can tell from the recipe, the parmesan cheese is included mostly for flavor, plus a little crispiness as suggested in a comment. The mozzarella and egg are what keep the base together.

Omitting the parmesan altogether should work. You can substitute any hard(er) cheese. I would also recommend breadcrumbs if cheese is too expensive, but that will probably not fit in the Keto diet.

As a final note: the recipe calls for 1/4 cup of grated parmesan. Given a quick search for the density of grated parmesan, this comes to about 25g (less than 1 oz.) of cheese. This does not seem like a lot to me, although you might disagree.