Preheat the oven to 400°F (200°C).
Cut the spaghetti squash in half lengthwise and scoop out the seeds.
Brush the inside of the squash halves with 1 tablespoon olive oil and season with 1/2 teaspoon salt and 1/4 teaspoon black pepper.
Place the squash halves cut side down on a baking sheet lined with parchment paper.
Roast the squash in the preheated oven for 35-40 minutes, or until the flesh is tender and can be shredded easily with a fork.
While the squash is roasting, heat the remaining 1 tablespoon olive oil in a medium skillet over medium heat.
Add the chopped onion and cook for 3-4 minutes until translucent.
Add the minced garlic, smoked paprika, dried oregano, crushed red pepper flakes, and cook for another 1 minute until fragrant.
In a large mixing bowl, combine the shredded cooked chicken, cooked onions and garlic mixture, Greek yogurt, shredded cheddar cheese, mozzarella cheese, Parmesan cheese, chicken broth, lemon juice, lemon zest, remaining 1/2 teaspoon salt, 1/4 teaspoon black pepper, and chopped fresh parsley. Mix well.
Remove the spaghetti squash from the oven and let it cool slightly.
Using a fork, shred the spaghetti squash flesh into strands and add it to the chicken and cheese mixture. Stir gently to combine.
Spray a 9×13-inch casserole dish with cooking spray.
Transfer the mixture into the casserole dish and spread evenly.
Bake the casserole at 375°F (190°C) for 15 minutes, or until the cheese is melted and bubbly.
Remove from the oven and let it rest for 5 minutes before serving.