Rinse the chicken well and pat completely dry. Season with salt and pepper. In a large (12-inch) skillet, heat the butter and olive oil over medium-high heat until the butter foams and subsides, then add the chicken and brown well on all sides, 10-12 minutes. Transfer chicken to plates and discard all but 2 tablespoons of oil from the pan.
Lower heat to medium and add the onion, garlic, and a pinch of salt and pepper. Cook, stirring often, until golden, 5-6 minutes.
Add rice and stir well to coat with fat, cooking for an additional minute. Add broth, water, saffron, and bay leaf, and bring to a boil, then turn down to a simmer. Taste for seasoning. Return chicken to pan, nestling into the rice until half-submerged. Pour in any juices from the plate and cover. Cook for 20-25 minutes, until the liquid is absorbed and the rice is nearly tender. If it is not yet tender, add boiling water as needed to keep moist.
Remove from the heat and stir in the peas. Allow to rest, covered with a kitchen towel, until the rice absorbs any remaining liquid and becomes fluffy. Discard bay leaf. Serve with parsley to garnish, if desired.