Heat oil in a 12-inch skillet over med.-high heat until very hot. Add chicken pieces and salt and cook until chicken is golden brown on all sides, transferring chicken to a plate as it browns. (I use a paper towel-lined plate to absorb the excess oil.).
Add onion to skillet and cook until golden, stirring frequently.
Stir in olives, tomatoes, capers, chicken broth, rosemary, lemon juice, and 1/3 cup water.
Return chicken to skillet and bring to boiling over high heat. Reduce heat to low, cover and simmer for 25-30 minutes. Spoon olive mixture over chicken and stir occasionally while chicken is cooking. Chicken is done when juices run clear when pieces are pierces with a knife.
To serve, arrange chicken on a platter, spoon olive mixture over chicken, and sprinkle with chopped parsley.