Heat the olive oil in a large skillet over medium-high heat. Add the chicken pieces and cook until browned on all sides and mostly cooked through, about 5-7 minutes. Season with salt and optional pepper.
Add the diced zucchini, yellow squash, and red bell pepper to the skillet. Stir well to combine with the chicken. Cook for another 5 minutes, stirring occasionally, until the vegetables begin to soften.
Stir in the broccoli florets, minced garlic (if using), dried thyme, and dried rosemary. Pour in the filtered water or low-histamine broth.
Cover the skillet and reduce the heat to medium-low. Let it simmer for 8-10 minutes, or until the broccoli is tender-crisp and the chicken is fully cooked through. Ensure the liquid has reduced slightly.
Remove the skillet from the heat. Stir in the fresh chopped parsley. Taste and adjust seasoning with salt if needed.
Serve immediately while hot.