Combine water and 2 tablespoons juice in a large bowl. Cut off stem from an artichoke to within 1 inch of base; peel stem. Remove bottom leaves and tough outer leaves, leaving tender heart and bottom. Cut lengthwise into quarters. Remove fuzzy thistle from bottom with a spoon; place in lemon water. Repeat procedure with remaining artichokes. Drain artichokes. Cook artichokes in boiling water 6 minutes or until tender. Remove from pan with a slotted spoon; plunge into a large bowl of ice water. Add carrots to pan; cook 1 minute. Add cauliflower and squash to pan; cook an additional minute or until vegetables are crisp-tender. Drain and plunge in ice water with artichokes. Drain artichoke mixture well.
Combine remaining 2 tablespoons juice, vinegar, and next 6 ingredients (through garlic) in a large bowl; stir with a whisk. Add artichoke mixture, onion, and olives; toss gently to coat. Cover and chill. Sprinkle with the parsley just before serving.