Put the flour, sugar, rosemary, salt, and Parmesan into the bowl of a food processor and pulse until combined. Add the butter and pulse just until a soft dough forms; the dough should hold together when squeezed with your hands. If not, add the water and pulse until combined.

Spread a large sheet of plastic wrap on a work surface and transfer the dough onto it. Using the plastic wrap as a guide, form the dough into a loose log along 1 edge of the long side of the sheet. Roll the dough log, twisting the plastic gathered at the ends in opposite directions until the log is tight and compact, about 2 1/2 inches in diameter. Chill in the refrigerator until firm, about 1 hour.

Preheat the oven to 375 degrees F. Line 2 baking sheets with parchment paper or silicone baking mats. Slice the dough log into 1/3-inch thick slices and arrange on the lined sheets, about 1-inch apart. Bake until the edges are just beginning to brown, 12 to 14 minutes.

Cool the shortbread on the pan for 5 minutes then transfer to wire racks to cool completely. Store the shortbread in an airtight container at room temperature until ready to serve.