Preheat the oven to 325 degrees F (165 degrees C). Grease a 2-quart casserole dish.
Heat oil in a large skillet over medium-high heat. Add celery and onion; cook and stir until tender, about 5 minutes.
Mix in bread crumbs and parsley and remove from the heat. Stir in oysters and eggs; season with salt, poultry seasoning, thyme, and pepper. Stir in enough liquid from the oysters to moisten and mix everything thoroughly.
Transfer to the prepared casserole dish.
Bake in the preheated oven until the top is toasted and a knife inserted into the center comes out clean, about 45 minutes.