Trim the pearl onions and cut a slit up the sides from root to tip. Transfer to a bowl and cover with hot water; let sit 30 minutes to soften, then drain and peel; set aside.
Meanwhile, wrap the thyme, bay leaves and juniper berries in a square of cheesecloth and tie closed; set aside.
Butterfly the brisket: With your knife parallel to the cutting board, slice the brisket in half horizontally almost all the way through, leaving one side attached; open like a book. (If the brisket is triangular, cut from the pointed side toward the wider side.) Pound with the flat side of a meat mallet until about 3/4 inch thick, if needed.
Season the brisket with salt and pepper and arrange with the wider of the two short sides in front of you. Arrange the carrots and parsnips horizontally across the brisket, close to the side in front of you. Starting from this side, roll the brisket over the vegetables to make a tight log and set seam-side down. Tie in 2-inch intervals using kitchen twine, then tie across the length, tucking in the ends. Season with salt and pepper.
Preheat the oven to 250 degrees F. Heat 2 tablespoons olive oil in a large Dutch oven over high heat. Add the brisket and cook, turning, until browned all over, 12 to 15 minutes. Remove to a plate.
Reduce the heat to medium. Add the remaining 1 tablespoon olive oil if the pot looks dry, then add the garlic, shallot and tomato paste and cook, stirring, until golden, about 1 minute. Sprinkle in the flour and cook, stirring constantly, until golden, about 1 minute. Add the wine, brandy, pearl onions and cheesecloth bundle and bring to a boil. Whisk in the demi-glace and 1 cup water and return to a boil.
Nestle the brisket in the liquid and return to a boil. Cover and transfer to the oven. Bake, turning halfway through, until the brisket is tender and a thermometer inserted into the center registers 190 degrees F to 200 degrees F, about 4 hours. Remove from the oven, uncover and turn the brisket. Let rest in the sauce at least 30 minutes.
Transfer the brisket to a cutting board. Discard the cheesecloth bundle. Skim the fat off the top of the sauce and cook over high heat until thick and glossy, 15 to 20 minutes. Remove from the heat and whisk in the butter, then stir in the parsley.
Remove the twine from the brisket and cut into 1-inch-thick slices. Transfer the sauce and brisket to a platter; top with more parsley.
Veal demi-glace is a thick stock that gives this sauce its rich flavor. It's sold refrigerated at butcher shops and specialty stores. If you can't find it, use 2 cups beef stock and omit the water in step 6.