Heat the oil in a large pot over medium heat. Add the garlic and cook until golden but not browned, about 1 minute. Add the tomatoes and their juices, red pepper, oregano, salt, and sugar. Bring to a boil. Reduce heat to medium-low and simmer (crushing the tomatoes with the back of a wooden spoon if using whole tomatoes) until the sauce thickens slightly, about 20 minutes. Add the mussels. Cover and cook until the mussels open, 4 to 6 minutes. Remove and discard any that don't open.
Meanwhile, cook the linguine according to the package directions. Add the linguine and the parsley to the sauce and toss.