Strain oysters, reserving their liquid. Place oysters in a large mixing bowl with the cubed bread.
Preheat oven to 400 degrees. Grease the inside of a 2-quart (or slightly larger) baking dish.
Finely chop celery and onion. Sauté in butter until tender. Mix in seasonings. Add oyster liquor and cook for 2-3 minutes. Remove from heat and cool to room temperature. Pour over oysters and bread. Mix gently until moistened.
Stir in the eggs and parsley until well distributed. Transfer to the baking dish; cover and bake for about 35 minutes, then remove the foil and bake for another 10 to 15 minutes, until top is browned.
Can be prepared 1-2 days in advance and kept refrigerated before baking.