Mix buttermilk, garlic, paprika, salt, and pepper together in a large bowl; stir to combine.
Massage rosemary sprigs to release fragrant oils. Put rosemary in a large resealable plastic bag; pour in buttermilk mixture. Add chicken parts to the bag, seal carefully, and place in refrigerator. Marinate, turning the bag occasionally, 4 hours to overnight.
Preheat grill for medium heat and lightly oil the grate.
Transfer chicken to a plate; discard marinade. Grill chicken until no longer pink at the bone and the juices run clear, turning a few times, 20 to 30 minutes. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C).