Season roast with 3 Tbsp. salt, rubbing into the grain and covering all sides. Wrap tightly in plastic and chill at least 3 hours and up to 3 days.
Heat 2 Tbsp. oil in a medium skillet over high. Cook onion, leek, and remaining 2 tsp. salt, tossing occasionally, until browned around the edges but not completely cooked through, 6–8 minutes. Add garlic and tomato paste and cook, tossing occasionally, until paste is brick red and mixture is fragrant, about 1 minute. Add wine and bay leaf and cook, stirring occasionally and scraping bottom of pan, until slightly reduced and you can no longer smell the alcohol, about 2 minutes. Scrape onion mixture into ceramic insert of a 6-qt. slow cooker. Stir in broth and pepper.
Heat remaining 2 Tbsp. oil in same skillet over high. Cook roast, turning occasionally, until golden brown on all sides, 10–12 minutes. Transfer to insert, nestling into onion mixture. Arrange carrots, celery, parsnip, potatoes, and mushrooms around roast, pushing into onion mixture and surrounding meat. You might think there isn’t enough liquid, but the juices from the roast and vegetables will release as they cook.
Slide ceramic insert into base, cover, and cook over high heat until roast is very tender and shreds easily, 7–8 hours. If programming your slow cooker, set for 7 hours at high, then set to warm after that.
Transfer roast to a cutting board and slice against the grain. Divide vegetables among plates. Arrange sliced roast over. Top with parsley and chives.