In a large pot, melt the butter over medium heat. Add the onion and celery and cook until the onion is translucent and the celery begins to soften.
Add the potatoes and carrots to the pot. Season with salt and pepper. Stir in the flour and cook for 1 to 2 minutes. Add the broth and bring to a boil. Reduce the heat to low and simmer gently until the potatoes are tender, about 45 minutes.
Remove from the heat and stir in the milk and parsley. Cook until the soup begins to bubble again. Remove from the heat.