Bring a large pot of salted water to a boil and cook the spaghetti according to the package directions until al dente. Drain and set aside.
In the same pot used for spaghetti, add the spiralized zucchini with 1/2 cup of water over medium-high heat. Cook until the zucchini is al dente, about 2-3 minutes. Drain and set aside.
Set a large skillet over medium heat. Add the olive oil and butter. Once the butter is melted, add the garlic and parsley. Cook for 1 minute, stirring often, or just until the garlic is lightly golden.
Reduce to low heat then add the cooked spaghetti, zucchini noodles, and chickpeas. Toss gently to coat everything evenly.
Add salt, black pepper, and red pepper flakes. Adjust seasoning to taste. Stir in the mozzarella balls, and serve warm.