To prepare artichokes, combine 2 cups water and 3 tablespoons lemon juice in a bowl. Cut off stem of each artichoke to within 1 inch of base; peel stem. Remove bottom leaves and tough outer leaves of each artichoke, leaving tender heart and bottom. Remove fuzzy thistle from bottom with a spoon. Cut each artichoke into quarters lengthwise. Place artichoke quarters in lemon water, and drain.
Combine 4 cups broth, 1/4 teaspoon salt, and 3 tablespoons lemon juice in a medium saucepan; bring to a boil. Add artichoke quarters; cook 20 minutes or until tender. Drain through a fine sieve over a bowl, reserving artichokes and 1/2 cup cooking liquid. Cover and set aside.
To prepare peas, combine 4 cups water, 2 cups broth, and peas in a medium saucepan; bring to a boil over medium-high heat. Reduce heat; simmer, partially covered, 30 minutes or until peas are tender. Drain.
Heat oil in a nonstick skillet over medium-high heat. Add shallots; sauté 3 minutes. Add garlic; sauté 1 minute. Add 1/2 cup reserved cooking liquid, artichokes, peas, 3/4 teaspoon salt, and pepper. Bring to a boil, and cook 1 minute. Stir in 3 tablespoons lemon juice and parsley.