Place the chicken breasts in a medium pot and cover with water. Bring to a boil over medium-high heat, then reduce heat to medium-low and simmer for 12-15 minutes or until the chicken is cooked through and no longer pink in the center.
Remove the chicken from the pot and let it cool slightly. Once cool enough to handle, finely chop the chicken into small pieces and transfer to a large mixing bowl.
Add the finely chopped celery, red onion, cucumber, red bell pepper, and parsley to the bowl with the chicken.
In a small bowl, whisk together the mayonnaise, Greek yogurt, lemon juice, Dijon mustard, salt, and black pepper until smooth and well combined.
Pour the dressing over the chopped chicken and vegetables. Stir gently until everything is evenly coated with the dressing.
Taste and adjust seasoning with additional salt and pepper if needed.
Cover the bowl with plastic wrap and refrigerate the chicken salad for at least 15 minutes before serving to allow the flavors to meld.
Serve chilled on its own, over a bed of lettuce, or as a sandwich filling.