Place black-eyed peas in a large pot and add enough water to cover by 2 inches. Bring to a boil over high heat. Remove from heat and let soak for 1 hour.
Drain black-eyed peas and return to same pot. Cut sausage into 3 pieces. Cut 2 carrots and 1 celery rib in half. Add cut sausage, carrots, and celery, plus parsley sprigs and bay leaves, to pot. Add enough water to cover by 2 inches. Bring black-eyed peas to a boil over high heat. Reduce heat to medium-low and simmer until tender, 45 minutes to 1 hour. Transfer sausage to a cutting board and peel away casing. Drain black-eyed peas (discard vegetables and herbs), transfer to a large serving bowl, and sprinkle with vinegar. Set aside.
Slice sausage into 1/2-inch-thick rounds; then cut each round into quarters. Dice remaining carrots into 1/4-inch pieces. In a large nonstick skillet over medium-high heat, heat 1 tablespoon olive oil. Add sausage and cook, stirring occasionally, until lightly browned, about 5 minutes. Add carrots and cook, stirring until softened, 3 to 4 more minutes. Transfer to a medium bowl.
Dice remaining celery into 1/4-inch pieces and add — along with remaining olive oil, onion, mustard, tarragon, paprika, and salt — to sausage mixture. Stir well to mix. Gently fold sausage mixture and remaining parsley into peas. Sprinkle with more paprika, as desired. Serve at room temperature or chilled.