Preheat the oven to 325 degrees. In a large heavy pot or Dutch oven, heat the oil over medium-high heat. Season both sides of brisket with 1 teaspoon each salt and pepper. When the oil shimmers, add the brisket fat side down and cook until dark golden, 3 to 4 minutes. Turn the brisket and cook the other side until browned, 3 minutes more. Remove the meat from pot and add onions, garlic, and thyme. Cook until the onions begin to soften and the ingredients become fragrant, about 4 minutes. Stir in tomato paste and 4 cups water and bring to a simmer. Return the brisket to the pot, fat side up; cover and transfer to the oven. Cook until almost tender, about 2 hours and 45 minutes.
Add the vegetables, submerging them slightly in the cooking liquid. Continue cooking until the vegetables are tender, 30 to 45 minutes more. Transfer the meat to a cutting board, and remove the vegetables with a slotted spoon. Skim as much of the fat from the cooking liquid as possible; season with additional salt and pepper. Slice the brisket against the grain. Serve, topped with parsley, with vegetables and cooking liquid on the side.