Add the beef brisket to the slow cooker. Layer the sliced onion and minced garlic on top, then pour in the beef broth and half the barbecue sauce. Sprinkle with smoked paprika, salt, and pepper. Cover and cook on low for 8 hours until the meat pulls apart easily.
When the brisket has about 45 minutes left, heat the oven to 400°F. Place the carrot chunks, parsnip chunks, and sweet potato cubes on a baking sheet. Drizzle with olive oil and season with salt and pepper, then toss to coat.
Roast the vegetables for 30 to 35 minutes, stirring once halfway through, until the edges turn golden and the centers are tender.
Remove the brisket from the slow cooker and shred it with two forks. Stir the shredded meat back into the cooking juices along with the remaining barbecue sauce.
Plate the brisket with the roasted root vegetables on the side.