Heat 2 Tbsp. oil in a medium pot over medium-high heat. Season chicken with salt; add to pot and cook, turning once, until browned, 8–10 minutes. Transfer to a plate. Reduce heat to low and let oil cool for 1 minute; add garlic and cook, stirring often, until fragrant, 30–60 seconds. Add cumin, tomato paste, and red pepper flakes; stir until a smooth paste forms, about 1 minute. Add reserved chicken with any accumulated juices, along with bay leaves and 4 cups water. Scrape up any browned bits. Bring to a boil; reduce heat to medium-low and simmer, uncovered, occasionally stirring, until chicken is tender, about 20 minutes.

Transfer chicken to a plate. Add chickpeas to pot; bring to a simmer and cook for 5 minutes. Shred chicken; add to stew. Add red peppers. Stir in remaining 2 Tbsp. oil and 2 Tbsp. lemon juice; simmer for 1 minute. Season with salt and more lemon juice, if desired. Divide bread cubes among bowls. Ladle soup over. Garnish with parsley.