Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat.
Add the chopped onion and sauté for 3-4 minutes until translucent.
Add the minced garlic and cook for 1 minute until fragrant.
Add the chopped cauliflower, mushrooms, and celery to the skillet. Cook for 5-7 minutes until the vegetables are tender, stirring occasionally.
Remove the skillet from heat and transfer the cooked vegetables to a large mixing bowl.
Add the drained tuna to the bowl with the vegetables and gently mix to combine.
In a separate small bowl, combine the sour cream, mayonnaise, Dijon mustard, dried thyme, salt, black pepper, and smoked paprika. Mix well.
Pour the sour cream mixture over the tuna and vegetables and stir until everything is evenly coated.
Add the shredded cheddar cheese and half of the mozzarella cheese to the mixture. Stir gently to combine.
Transfer the mixture to a greased 8×8 inch baking dish and spread evenly.
Sprinkle the remaining mozzarella cheese and grated Parmesan cheese evenly over the top of the casserole.
Bake in the preheated oven for 20-25 minutes, or until the cheese is melted and bubbly and the top is golden brown.
Remove from the oven and let the casserole cool for 5 minutes.
Sprinkle chopped fresh parsley over the top before serving.