Preheat oven to 350°. In a large pot of boiling salted water, cook spaghetti, stirring occasionally, until al dente according to package instructions, about 7 minutes.
Meanwhile, in another large pot over medium heat, heat oil. Add onion and bell pepper and cook, stirring occasionally, until softened, about 7 minutes. Transfer to a plate.
Season chicken all over with salt, pepper, and chili powder. In same pot over medium-high heat, cook chicken, stirring occasionally, until golden brown, about 5 minutes. Transfer to another plate.
In same pot over medium heat, melt butter. Whisk in flour and cook, whisking constantly, until fragrant and combined, about 1 minute. Add milk and broth and cook, stirring occasionally, until thickened, about 5 minutes.
Stir in spaghetti, vegetables, chicken, and 1 ½ cups cheddar. Transfer to a 13" x 9" baking dish. Top with remaining 1 cup cheese.
Bake chicken spaghetti until bubbly, about 45 minutes. Top with parsley.
