Coat the brisket on all sides with the rub and actually rub it in. Pull out a length of plastic wrap for the brisket and drizzle some oil where you plan to place the brisket. Place the brisket on top and drizzle the top with oil. Wrap tightly and rest at room temperature for 2 hours. If you are making this ahead of time, you can refrigerate, then rest on the counter for 2 hours before braising.
Add 2 tablespoons of oil to a large Dutch oven over medium-high heat. When the oil begins to swirl, add the brisket and sear on all sides until browned, 3 to 4 minutes per side. Remove to a plate.
Add the onions, carrots and potatoes to the pot and cook for a few minutes to coat them in the fat and get a bit of color. Make the contents of the pot as level as possible, toss in the garlic, then place the seared brisket on top of the vegetables.
Stir together the BBQ sauce and stock in a medium bowl. Add to the pot around the brisket, being careful to only fill halfway up the brisket, leaving a bit of the seared meat above the liquid line (you may not use all the liquid). Raise to a boil and lower to a low simmer. Cook, covered, for 2 to 3 hours without peeking, until the brisket is tender!
Remove the brisket from the pot to a plate and cover gently with aluminum foil. Remove the vegetables from the pot with a slotted spoon and place in a food processor. Blitz until smooth, adding liquid from the pot if needed for texture. Use the remaining liquid as a sauce for the brisket, if desired. Transfer the mash to a bowl and fold in the parsley. Serve with the sliced brisket and additional sauce.