Preheat oven to 400 degrees with oven racks in upper and lower thirds of oven. Lightly oil 2 large baking sheets.
Divide pizza dough in half. Keep half of dough covered with a clean kitchen towel. Gently roll out other half into a 10" square on a lightly floured surface with a lightly floured rolling pin. Use hand to pull corners, if dough is very elastic. Cover with a clean kitchen towel and allow to rest about 3 minutes.
Cut dough square in half with a pizza wheel or a sharp heavy knife, then cut each half crosswise into 15 strips (about 2/3" wide). Cover strips with clean kitchen towel.
Keeping remaining strips covered, gently tie each strip into a knot, pulling ends slightly to secure (if dough is sticky, dust lightly with flour) and arranging knot 1" apart in staggered rows on 1 large baking sheet. Keep knots covered with clean kitchen towel.
Roll out and cut remaining dough, then form into knots and arrange on second baking sheet.
Bake, switching position of baking sheets halfway through baking, until golden, 20-25 minutes.
While knots bake, mince garlic and mash to a paste with salt, then stir together with oil in a very large bowl. Immediately after baking, toss knots in garlic oil, then sprinkle with parsley and cheese and toss to coat. Serve warm or at room temperature.
Knots can be baked, but not coated 2 hours ahead. Reheat on a large baking sheet in a preheated 350 degree oven until hot, 3-5 minutes. Toss immediately with garlic oil and sprinkle with cheese and parsley.