Preheat oven to 400°.
Heat olive oil in a 2-quart ovenproof saucepan. Add mushrooms and onion; cook 5 minutes. Add rice, broth, salt, and saffron; bring to a boil. Wrap handle of pan with foil; cover and transfer pan to oven. Bake at 400° for 15 minutes. Remove from oven; let stand, covered, 10 minutes. Add parsley and thyme.