In a medium Dutch oven or soup pot, combine the chicken broth, carrots, celery, onion halves, garlic cloves, and parmesan rind. Bring to a rolling boil over medium-high heat.

Reduce heat to medium-low, cover partially with a lid, and simmer gently until the carrots and celery are completely fork-tender.

Remove and discard the parmesan rind. Using a slotted spoon, transfer the softened carrots, celery, onion, and garlic to a blender with one cup of the warm broth. Puree until completely smooth, then pour the mixture back into the soup pot and stir to combine.

Bring the velvety broth back to a gentle simmer. Stir in the pastina and cook, stirring frequently to prevent sticking, until the pasta is tender and cooked through.

Turn off the heat. Vigorously stir in the butter and freshly grated Parmigiano-Reggiano until melted and creamy. Season with salt and black pepper to taste, then ladle into warm bowls and garnish with chopped parsley.