Cooking pizzas on the grill is easy and fun for the whole family.
Build a medium-hot fire in an outdoor grill, or set grill pan on stove over medium-high heat. Divide pizza dough in half. Working on a clean baking sheet greased with 1/2 tablespoon olive oil, use your fingers to flatten and stretch the dough into a rough 10-inch circle or rectangle. Let rest a few minutes; then carefully stretch again, as the dough will shrink. The dough should be as thin as possible without tearing.
Transfer dough to grill, and cook until bottom side is golden brown and top side starts to bubble, 3 to 5 minutes. quickly turn crust over, and cook about 2 minutes more to ensure that crust cooks through. Turn the crust over again. Scatter half the tomatoes and half the mozzarella on top of the crust; sprinkle with half the parsley. Cover grill, and let cook until toppings are hot and bubbly, 6 to 8 minutes. Remove from grill; slide pizza onto a cutting board. Drizzle with olive oil; season with salt and pepper. Repeat process with remaining dough and toppings. Cut, and serve.
Cutting toppings into small pieces helps them to cook faster and more evenly.