Heat the oil in a large heavy pot over medium-high heat until hot. Add the garlic, onions, 1 teaspoon salt and 3/4 teaspoon pepper and cook, stirring occasionally, until golden, about 10 minutes. Stir in the wine, pepper flakes and thyme, and boil until reduced by half, about 1 1/2 minutes. Stir in the pureed tomatoes and broth and bring to a simmer. Add the mussels and cover the pot. Cook until the mussels have opened wide, about 6 minutes. Discard any that didn't open. Season with salt and pepper.
Transfer to serving bowls and sprinkle with the parsley. Serve with crusty bread.