Place a rack in lowest position of oven; preheat to 350°. Tie 1 small bunch thyme, 5 sprigs parsley, and 2 bay leaves together with kitchen twine; set aside. Sprinkle one 4-lb. piece untrimmed flat-cut beef brisket, excess fat trimmed, all over with 4 tsp. Diamond Crystal or 2½ tsp. Morton kosher salt; season generously with freshly ground pepper.
Heat 2 Tbsp. vegetable oil in a large Dutch oven or other heavy pot over medium-high. Cook brisket until well browned, about 5 minutes per side. Transfer to a platter. Pour fat in pot into a small bowl, then add 1 Tbsp. back to pot; set remaining fat aside.
Cook 4 oz. bacon (about 4 slices), coarsely chopped, in same pot, stirring occasionally, until browned and crisp, about 4 minutes. Using a slotted spoon, transfer bacon to a medium bowl. Pour off fat into bowl with reserved fat; add 2 Tbsp. back to pot.
Cook 8 oz. crimini mushrooms, halved, quartered if large, 1 large carrot, peeled, coarsely chopped, 2 cups frozen pearl onions (from a 14.4-oz. bag), thawed, and 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt in pot, stirring occasionally, until vegetables are softened and browned, 6–8 minutes. Transfer to bowl with bacon; set aside.
Add 2 Tbsp. fat to pot. Cook 5 garlic cloves, smashed, stirring often, 30 seconds. Stir in 2 Tbsp. all-purpose flour and cook, stirring constantly, 30 seconds. Add 2 Tbsp. tomato paste and cook, stirring often, until slightly darkened in color, about 2 minutes. Pour in 3 cups dry red wine, scraping up any browned bits stuck to the bottom of pot with a wooden spoon. Bring to a boil and cook until wine is reduced by a third, about 2 minutes. Remove from heat and stir in 3½ cups low-sodium beef broth and remaining 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt. Carefully return brisket to pot, arranging fat side up; tuck reserved herb bundle alongside.
Cover pot and transfer to oven. Braise brisket, spooning braising liquid over a few times, 3½ hours. Uncover pot, remove and discard herb bundle, and stir in reserved bacon and vegetables. Return pot to oven and cook, uncovered, until brisket is deeply browned and very tender, 30–40 minutes. Let cool slightly.
Transfer brisket to a cutting board and slice against the grain. Skim fat from surface of braising liquid if needed, then return brisket to pot. Top with chopped parsley leaves just before serving.
Do Ahead: Brisket Bourguignon (without parsley) can be made 1 day ahead. Let cool completely; cover and chill. Reheat, covered, over medium heat until brisket is warmed through, 12–18 minutes.