Working with 1 artichoke at a time, cut off top inch of artichoke and gently pull open center. Scoop out sharp leaves and fuzzy choke from center with a melon-ball cutter or a spoon. Trim bottom 1/4 inch of stem (if present), keeping stem attached, and peel stem. Put artichoke in a large bowl of cold water. Repeat with remaining artichokes.
Remove artichokes from water and push 3 parsley sprigs into center of each. Heat 1/4 cup olive oil in pot over moderate heat until hot but not smoking, then add artichokes, thyme sprigs, garlic, 1/4 cup water, pepper, and 1/2 teaspoon salt. Cover pot and braise artichokes, turning occasionally, until artichokes are browned in spots and bases are tender when pierced with a knife, about 35 minutes.
Transfer artichokes, thyme, and garlic to a platter. Add remaining 3/4 cup water to pot and deglaze by boiling over high heat, stirring and scraping up brown bits, 1 minute. Pour pan juices (they will be dark) into a small bowl and stir in extra-virgin olive oil and remaining 1/2 teaspoon salt. Squeeze pulp from 2 of garlic cloves into juices and mash into sauce with a fork.
Divide artichokes and remaining garlic cloves among 6 plates and drizzle with sauce. Garlic cloves can be peeled and spread on crusty bread.