Bring the broth to a simmer in a 2 to 3 quart saucepan. Add chicken and simmer, uncovered, 6 minutes. Remove pan from heat and cover, then let stand until chicken is cooked through, about 15 minutes. Transfer chicken to a plate and cool 10 minutes. Reserve poaching liquid, uncovered.
While chicken is poaching, cook onion in oil in a 4 quart heavy pot, covered, over moderate heat, stirring occasionally, until softened but not browned, about 6 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute. Add carrots, celery, salt, and pepper and cook, covered, stirring occasionally, until softened, 8 to 10 minutes. Add poaching liquid and simmer, covered, until vegetables are tender, about 10 minutes. Remove from heat.
While vegetables are cooking, shred chicken into 1/4-inch wide strips (about 1-inch long). When vegetables are done simmering, stir chicken into soup along with parsley.