In a large pot, bring 2 quarts water to a boil. Add the pasta and 1 tablespoon salt, then cook, stirring occasionally, until just shy of al dente. Reserve ½ cup of the cooking water, then drain; set aside.
Meanwhile, in a 12-inch skillet over medium-high, combine the oil and garlic; cook, stirring often, until the garlic is lightly golden, 1 to 2 minutes. Add the oregano and pepper flakes, then cook, stirring, until fragrant, about 30 seconds. Add the tomato juices and Parmesan rind (if using). Bring to a simmer and cook, stirring often, until a spatula leaves a trail when drawn through the mixture, about 5 minutes.
Add the crushed tomatoes and cook, stirring often, until heated through, about 2 minutes. Add the pasta and ¼ cup of the reserved cooking water, then reduce to medium and cook, stirring and tossing, until the pasta is al dente and lightly sauced, about 3 minutes; add more cooking water 1 tablespoon at a time as needed if the mixture looks dry.
Off heat, remove and discard the garlic cloves, Parmesan rind (if used) and oregano sprigs (if used). Taste and season with salt and black pepper. Serve drizzled with additional oil and sprinkled with Parmesan.
