1. Preheat the oven to 350°F.
2. Cut the peppers in half and place them skin side down on a baking sheet. Bake for 25 minutes, until the skin is charred.
3. Remove them from the oven and let them cool; peel the peppers by pulling the skin off in one piece. Slice them into strips.
4. In a pot of boiling salted water, cook the Jerusalem artichokes for 15 minutes. Drain the cooking water and cool them in ice water.
5. Drain the Jerusalem artichokes and transfer them to a bowl with the roasted peppers.
6. Soak the dried tomatoes in boiling water for 5 minutes.
7. Cover and let them rest. Drain.
8. In a food processor, puree the soaked tomatoes. Transfer them to a small bowl, add the vinegar, and slowly pour in the olive oil. Season with salt and pepper to taste, then mix with the previous vegetables and add the parsley.
9. Serve as a salad.