Preheat the oven to 400°F (200°C).
Cut the spaghetti squash in half lengthwise and scoop out the seeds.
Brush the inside of each half 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 oven for 35-40 minutes, or until the flesh is tender and can be shredded with a fork.
While the squash roasts, season the chicken breasts with 1/2 teaspoon salt, 1/4 teaspoon black pepper, garlic powder, and smoked paprika.
Heat 1 tablespoon olive oil in a large skillet over medium heat.
Add the chicken breasts and cook for 6-7 minutes on each side, or until fully cooked and no longer pink inside.
Remove the chicken from the skillet and let rest for 5 minutes, then shred it using two forks.
In the same skillet, add the chopped onion and cook for 3-4 minutes until translucent.
Add the minced garlic and cook for 1 minute until fragrant.
Pour in the chicken broth and bring to a simmer, scraping any browned bits from the bottom of the pan.
Reduce heat to low and stir in the Greek yogurt, lemon juice, and lemon zest until smooth.
Add the shredded cheddar cheese, mozzarella cheese, and half of the Parmesan cheese to the sauce, stirring until melted and combined.
Remove the spaghetti squash from the oven and use a fork to shred the flesh into strands, transferring it to a large mixing bowl.
Add the shredded chicken and cheese sauce to the spaghetti squash and mix thoroughly.
Transfer the mixture to a greased 9×13-inch casserole dish.
Sprinkle the remaining Parmesan cheese evenly over the top.
Bake the casserole at 375°F (190°C) for 15 minutes, or until bubbly and golden on top.
Remove from the oven and garnish with chopped fresh parsley before serving.