Heat the oil in a large saucepan over medium heat. Add the carrot, celery, onion, measured salt, measured pepper, and thyme to the saucepan and stir to combine. Cook, stirring occasionally and scraping up any browned bits from the bottom of the pan, until the vegetables have softened and are just starting to brown, about 5 minutes.
Add the broth or stock and stir to combine. Add the bay leaf. Increase the heat to medium high and bring to a boil. Add canned chicken. Reduce the heat to medium low and simmer, about 20 minutes. Turn off the heat.
Break the noodle block into large pieces, add them to the saucepan, and stir to combine. Let sit, stirring occasionally, until the noodles are cooked through, about 5 minutes. Add the parsley and stir to combine. Taste and season with salt and pepper as needed.