Stir 1 Tbsp. salt and pasta into 3 qt. boiling water in a Dutch oven. Cook 3 minutes less than package directions state. Stir in broccoli florets; cook 3 minutes or until pasta is al dente, and broccoli is crisp-tender. Drain and return pasta and broccoli to Dutch oven.
While pasta cooks, combine olive oil and next 3 ingredients in a small saucepan. Cook over medium-low heat 6 minutes or until garlic is golden, stirring often. Remove from heat. Add garlic oil and parsley to pasta, and toss well. Serve hot.