Step 1Bring a small saucepan of water to boil over high heat. Cut the tops off the peppers and remove any seeds. Cut the peppers into half-inch wide strips and set aside.
Step 2In a medium (4-quart) saucepan, stir together the onion, garlic, carrot and olive oil until the vegetables are coated with oil. Cook over medium heat until the onion has softened and is fragrant, about 5 minutes.
Step 3Add the peppers, tomato pulp and parsley to the hot onion mixture. Pour three-fourths cup of boiling water over the mixture and reduce the heat to maintain a simmer. Cook until the peppers are floppy and almost limp, about 30 to 35 minutes, stirring occasionally. If the mixture cooks dry in the bottom of the pan, add a little more boiling water.
Step 4Transfer the mixture to a serving bowl and let cool to room temperature. Just before serving, stir in the salt and pepper and then balsamic vinegar to taste (1 teaspoon of traditional balsamic should be plenty).