In a large pot of boiling salted water, cook pasta according to package instructions. Reserve 1/2 cup pasta water. Drain pasta and return to pot.
Meanwhile, in a large skillet, heat 2 teaspoons oil over medium-high. Pat chicken dry with paper towels and season with salt and pepper. Cook chicken until golden and cooked through, about 2 minutes per side, flipping once. Transfer to a plate and let rest 5 minutes. with two forks, shred chicken.
Wipe out skillet and heat 2 teaspoons oil over medium-high. Add bell peppers and onion, season with salt and pepper, and cook, stirring occasionally, until vegetables are lightly browned and softened, about 15 minutes. Stir in parsley. Add vegetables and chicken to pasta. Toss, adding enough pasta water to coat pasta, and season with salt and pepper. To serve drizzle with oil.