Preheat oven to 400°F. Brush squash halves with 1 tbsp olive oil, season with salt and pepper, and roast cut-side down 30–40 minutes until strands separate.
Pat chicken dry, rub with Cajun seasoning and remaining olive oil. Sear in a hot skillet 4–5 minutes per side until cooked through; rest 5 minutes and slice.
Melt butter in same skillet, add garlic and sauté 30 seconds until fragrant. Stir in broth and lemon juice, scraping browned bits.
Use a fork to fluff spaghetti squash into strands, toss with garlic butter, then top with sliced Cajun chicken and parsley.