Bibb is a type of butterhead lettuce, with tender, sweet leaves. If you can’t find it, substitute Boston, another butterhead variety.
Combine parsley stems, oil, and thyme sprigs in a small saucepan. Cook over medium heat, stirring occasionally, for 3 minutes. Remove from heat; let stand 10 minutes. Pour mixture through a fine wire-mesh strainer into a large bowl, pressing to release oil. Discard solids. Whisk in vinegar.
Cook pasta according to package directions, omitting salt and fat. Add peas during last 2 minutes of cooking; drain. Let stand 5 minutes.
Add pasta mixture, shallot, salt, and pepper to oil mixture; toss to coat. Add lettuce, chopped parsley leaves, and chives; stir gently to combine. Sprinkle with cheese.