SEASON the chicken with the oregano, ½ teaspoon of the salt, and the pepper. Add the coconut oil to a 6-quart Instant Pot. Select Sauté and adjust to Normal/Medium. When the oil is hot, add half of the chicken and cook, turning once, until browned on both sides, 4 to 8 minutes. Repeat with the remaining chicken. Select Cancel. Transfer the chicken to a plate.
ADD the onion, bell pepper, mushrooms, tomatoes, broth, tomato paste, garlic, and remaining ½ teaspoon salt to the pot. Add the chicken. Lock the lid in place.
SELECT Manual and cook on high pressure for 12 minutes. Use quick release.
TRANSFER the chicken to a plate; cover to keep warm. Select Sauté and adjust to Normal/Medium. When the sauce is simmering, add the zucchini noodles. Cook, stirring frequently until the sauce is thickened and the zucchini is crisp-tender, about 2 minutes. Select Cancel. Serve, topped with parsley if desired.