In 5- to 6-quart saucepan or Dutch oven, heat olive oil on medium until hot. Add onion and cook about 8 minutes or until beginning to brown, stirring occasionally. Add garlic; cook 1 minute. Stir in wine and saffron; heat to boiling on medium-high.
Add mussels to wine mixture in saucepan and cook, covered, 5 to 7 minutes or until shells open, shaking closed pot occasionally. Discard any unopened mussels.
To serve, ladle mussels with their broth into 6 large soup bowls; sprinkle with parsley. Pass around warmed bread to dip in broth.