Warm olive oil in a large heavy-bottomed pot over medium-high heat. Sauté carrots, onion, and celery until softened, 3 to 5 minutes. Stir in garlic and thyme and cook for 30 more seconds.
Stir in chicken broth, 1 teaspoon salt, and pepper and bring to a boil over high heat. Add chicken and bay leaf, reduce heat to medium, and cover. Cook for 5 minutes, then stir in the tortellini, recover, and continue cooking.
Cook for 5 more minutes, then transfer the chicken to a bowl and use two forks to shred. Return chicken to the pot. If you used fresh tortellini, it should be ready to eat. If you used dried, you may need to continue cooking for another couple of minutes.
Stir spinach and 1/4 cup parsley into the soup, and combine the remaining 1/4 cup parsley with the zest of 1 of the lemons and 1/4 teaspoon salt. Cut the two lemons in half and squeeze the juice into the soup. Taste and add more salt or lemon juice if desired.
Serve soup with parsley-lemon zest mixture and fresh parmesan shavings.