Heat a large skillet over medium-high heat.
Sprinkle the chicken with salt and pepper. Sear the chicken in the olive oil on both sides until golden brown, about 4 minutes per side. Remove to a plate.
Add the anchovies and garlic to the hot pan, using your wooden spoon to break up the anchovy, and saute until the garlic is fragrant and the anchovy is melted. Add the onions, carrots and celery, and saute until tender and just beginning to turn golden. Add the white wine and stir well until the sharp smell of the wine is cooked off, about 2 minutes. Add the tomatoes and their juices along with the olives, and give a good stir. Add the chicken pieces to the skillet, making sure they are just covered by the tomatoes, flipping to coat with the sauce. Cover with a lid and simmer on low for 35 to 40 minutes. Serve with the chopped parsley.