Heat the vegetable oil in a large Dutch oven over medium-high heat. Add the lamb cubes and brown on all sides, about 6 to 8 minutes. Remove the lamb and set it aside.
Add the onion to the same pot and cook until softened, about 4 minutes. Stir in the garlic, ginger, and bruised lemongrass stalks. Cook for 1 minute until fragrant.
Stir in the Thai red curry paste and cook for another minute. Return the lamb to the pot. Pour in the coconut milk and chicken broth. Bring the mixture to a simmer.
Add the carrots, parsnips, and butternut squash. Cover and simmer gently for 45 to 50 minutes until the lamb is tender and the vegetables are cooked through.
Remove the lemongrass stalks. Stir in the fish sauce and lime juice. Taste and adjust with salt and pepper if needed. Serve hot, topped with chopped cilantro and lime wedges on the side.