Shell the peas and the fava beans and set them aside in separate containers. Carefully slice off the tips of the artichokes, remove all of the tough outer leaves, and peel the base. Cut the artichoke into quarters and remove the choke, as well as any prickly tips that may have formed on the inner leaves. Cut each of these quarters into thirds, so as to have tiny wedges. Let artichokes sit in water and lemon juice.
Fill a heavy-bottomed saucepan with 1/4 cup hot water. Slice the scallions and add them to the water. Cook over medium heat, stirring constantly, until all of the water has evaporated. Add the 4 tablespoons olive oil and saute the scallions until a pale golden, about 3 minutes.
Add the artichokes, 1/4 cup hot water, bay leaf and salt, to taste. Cover and cook over low heat until the artichokes begin to soften, about 5 minutes.
Add the fava beans and salt, to taste. Cover and cook another 5 minutes.
Add the peas and 2 tablespoons olive oil. Cover and cook for 5 more minutes, sprinkle with the vinegar, and add some black pepper, to taste. Continue cooking, uncovered over medium-high heat for until the vinegar has evaporated, about 2 to 3 minutes. Refrigerate for 24 hours and serve cold drizzled with remaining extra-virgin olive oil and sprinkled with parsley. This dish can be served warm, if desired.