Preheat the oven to 425 degrees F (220 degrees C). Lightly oil 2 baking sheets.
Roll 1 package of pizza dough out into a 10-inch square on a lightly floured surface using a lightly floured rolling pin, using hands to pull corners. Cut square in half using a pizza wheel or sharp knife, then cut each half into 15 strips about 2/3 inch wide. Dust dough lightly with flour if necessary to prevent sticking. Gently tie each strip into a knot, pulling ends slightly to secure. Arrange knots 1 inch apart on a prepared baking sheet. Repeat with remaining dough.
Bake in the preheated oven for 15 minutes.
Meanwhile, combine oil, garlic, and salt in a large bowl.
Remove knots from the oven and brush with garlic oil. Return knots to the oven and continue baking until golden brown, 5 to 10 minutes more.
Remove knots from oven and immediately sprinkle with Parmesan cheese and parsley. Serve warm.