Grease a 4- to 6-quart slow cooker with 1 tablespoon of the olive oil.

Cut 5 slits into each of the chicken breasts, making sure not to cut through the chicken. Stuff each slit with mozzarella and tomato slices. Place the stuffed chicken breasts into the slow cooker.

Pour the remaining 2 tablespoons olive oil over the stuffed chicken breasts. Sprinkle sea salt and black pepper over the chicken. Cover and cook on LOW for 6 hours or until a meat thermometer inserted in the thickest part of the chicken registers 165°F.

Serve topped with the parsley.