For sauce, in a large skillet cook sweet peppers and garlic in oil over medium heat about 20 minutes, stirring occasionally. (Or, if using peppers from a jar, in a 2-quart saucepan cook garlic in hot oil for 3 to 4 minutes or until light brown.)
Place half of the pepper-garlic mixture in a blender container or food processor bowl. Cover; blend or process until nearly smooth. Add half each of the water, basil, tomato paste, and vinegar. Cover and blend or process with several on-and-off turns until basil is just chopped and mixture is nearly smooth. Transfer to a 2-quart saucepan. Repeat with remaining peppers, water, basil, tomato paste, and vinegar; transfer to the saucepan.
Cook and stir sauce over medium heat until heated through. Serve sauce over pasta. If desired, sprinkle with Parmesan cheese.