Using a spiralizer, mandoline, or vegetable peeler, turn the zucchini into thin noodles. Microwave the noodles for 1-2 minutes, then drain any liquid.
Melt the butter in a large pan over medium heat. Add the chicken, salt, pepper, and garlic and cook until the garlic is beginning to brown and chicken is cooked through, about 7 minutes. Remove the chicken from the pan.
Add the cream, Parmesan, and parsley to the pan, stirring until evenly combined. Bring to a boil, then stir until the sauce has reduced by about half, 3-5 minutes. Put the chicken back in the sauce and stir to coat. Remove from the heat and season to taste with salt and pepper.
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Bring to a boil, then stir until the sauce has reduced by about half. Remove from heat.
Toss the zucchini noodles in with the chicken mixture and stir until the noodles are coated evenly. Serve with more Parmesan and parsley, if desired.