In a medium saucepan, simmer the chicken and onion in lightly salted water until the chicken is cooked. Remove the chicken, cool for 5 minutes, and shred into a medium bowl. Discard the onion. Place the chicken in a large bowl and refrigerate for 1 hour or until chilled.
Add the potatoes, carrot, peas, parsley, and mayonnaise to the chicken and mix gently to combine. Season with salt and pepper and set aside.
Fill each avocado half with the chicken mixture and serve.