In a large pot over high heat, add the turkey parts and cover with 4 quarts water. Add the thyme, bay leaves, and a big pinch of salt and bring to a boil. Reduce the heat to simmer, skimming any scum that rises to the surface. Cook until the turkey is cooked through and the liquid has reduced a bit, about 45 minutes. Remove the turkey from the pot. When it is cool enough to handle, shred the meat and discard the skin and bones. Strain the broth.
Wipe out the pot, put it over medium-high heat, and add the oil. When it is hot, add the onions, celery, carrot, and parsnip, season with salt and pepper, and cook until softened, 7 to 8 minutes. Add the garlic and cook for another minute. Add the turkey broth and bring to a simmer. Add the meat to the pot and let it simmer to warm through, about 10 minutes. Remove from the heat and stir in the parsley just before serving.
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