Heat 1 T. oil in Dutch oven over medium-high heat until shimmering.
Add onion, dark green leek part and celery, and cook until just beginning to brown, about 5 minutes.
Add chicken skin and bones, broth, water, thyme and bay leaves and bring to boil. Reduce heat to medium-low and simmer, loosely covered, for 30 minutes.
Strain stock through fine-mesh strainer into large bowl, pressing on solids to extract as much liquid as possible; discard solids and set stock aside. (You should have about 8 cups of stock. If you have less, add water to equal 8 cups.)
Wipe out now empty Dutch oven with paper towels and heat remaining 1 T. oil over medium-high heat until shimmering.
Add white and light green leek parts and carrots and cook until softened, about 5 minutes.
Add rice and cook stirring frequently, until edges of rice become translucent, about 2 minutes.
Add stock and 1 tsp. salt and bring to boil. Reduce heat to medium-low and simmer, covered, stirring occasionally, until rice is tender, 12 to 14 minutes.
Stir in chicken and peas and cook until heated through, about 5 minutes.
Stir in parsley and season with salt and pepper to taste. Serve.