Preheat the oven to 400°F (200°C).
Cut the spaghetti squash in half lengthwise and scoop out the seeds.
Drizzle 1 tablespoon olive oil over the cut sides of the squash 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 and roast for 40 minutes or until the flesh is tender and can be shredded with a fork.
While the squash roasts, heat 1 tablespoon olive oil in a large skillet over medium heat.
Season the chicken breasts with 1/2 teaspoon salt, 1/4 teaspoon black pepper, garlic powder, and smoked paprika.
Cook the chicken breasts for 5-7 minutes per side until fully cooked and no longer pink inside.
Remove the chicken from the skillet and let it rest for 5 minutes, then shred it using two forks.
In the same skillet, add the diced onion and cook over medium heat for 4-5 minutes until translucent.
Add the minced garlic and cook for an additional 1 minute until fragrant.
Melt the butter in a medium saucepan over medium heat.
Whisk in the flour and cook for 1-2 minutes until lightly golden, stirring constantly.
Gradually whisk in the chicken broth and milk, continuing to whisk until the sauce thickens, about 3-4 minutes.
Remove the sauce from heat and stir in the Greek yogurt, shredded cheddar, mozzarella, and Parmesan cheese until smooth and creamy.
Add the chopped jar of lemons to the cheese sauce and stir to combine.
Once the spaghetti squash is cool enough to handle, use a fork to shred the flesh into strands and transfer it to a large mixing bowl.
Add the shredded chicken, cooked onions and garlic, and cheese sauce to the spaghetti squash. Mix well to combine.
Transfer the mixture to a greased 9×13-inch casserole dish and spread evenly.
Bake the casserole at 375°F (190°C) for 15 minutes until bubbly and golden on top.
Remove from the oven and sprinkle with chopped fresh parsley before serving.