Pat chicken dry with paper towels and season with salt and pepper.
Heat 1 tablespoon oil in 12-inch skillet over medium-high heat until just smoking.
Brown half of chicken, about 4 minutes per side; transfer to slow cooker.
Repeat with 1 tablespoon oil and remaining chicken; transfer to slow cooker.
Heat remaining 1 tablespoon oil in now-empty skillet over medium heat until shimmering.
Add onions and cook until softened and lightly browned, 8 to 10 minutes.
Stir in flour, garlic, tomato paste, and thyme and cook until fragrant, about 1 minute.
Slowly whisk in 2 cups broth and wine, scraping up any browned bits and smoothing out any lumps.
Stir in potatoes and carrots and bring to simmer.
Cover, reduce heat to medium-low, and simmer until vegetables just begin to soften, about 10 minutes; transfer to slow cooker.
Stir remaining 2 cups broth and bay leaves into slow cooker, cover, and cook until chicken is tender, 4 to 5 hours on low.
Transfer chicken to cutting board, let cool slightly, then pull apart into large chunks using 2 forks.
Discard bay leaves.
Stir chicken and peas into stew and let sit until heated through, about 5 minutes.
Adjust consistency with extra hot broth as needed.
Stir in parsley and season with salt and pepper to taste.
Serve.
