Preheat an oven to 350 degrees F.
In a large pot, heat the canola oil over high heat. When the oil begins to smoke slightly, season both sides of the brisket with salt and pepper, to taste. Use metal tongs to add the meat to the pot. If the oil is hot enough, the meat will not stick to the bottom. Cook, undisturbed, on its first side until browned, about 3 to 5 minutes. Turn over and brown on the other side, about 3 to 5 additional minutes. Remove the meat from the pot and put it onto a baking sheet to rest.
Combine the garlic paste and green onions in a small bowl. Spread half of the mixture over the brisket. Flip the meat and add the remaining mixture to the other side. Set aside.
In the same pot, over low heat and add the parsnips. Season them with a little salt and a scant 1 tablespoon brown sugar. Stir to coat with the oil and cook until slightly tender, about 2 to 3 minutes. Add in 2 tablespoons apple cider vinegar and the leeks and gently brown them, about 3 to 5 minutes. Adjust the seasonings with salt. Pour in the wine and let it reduce for 5 minutes. Sprinkle the vegetables with the remaining 1/4 cup of brown sugar and another pinch of salt. Cook them until browned, another 3 to 5 minutes.
Add the brisket to the pot and stir in 2 to 3 cups of the beef stock. Bring the stock to a gentle simmer. Slide it into the center of the oven and cook for 1 hour. Check the level of the liquid, adding water, if needed. Remove the pot from the oven, taste and re-season, if needed. Return the pot to the oven and cook the brisket until the meat is tender when pierced with a fork, an additional 1 to 1 1/2 hours (see Cook's Note).
Remove the pot from the oven and transfer the meat to a platter. Arrange the vegetables alongside and ladle the sauce on top.
If the meat looks dry or begins to overly brown, cover the pot with a lid or a layer of aluminum foil for the remainder of the cooking process.