Pat the drumsticks dry with paper towels. In a small bowl, combine paprika, garlic powder, onion powder, salt, and pepper. Drizzle the drumsticks with olive oil and coat generously with the seasoning mixture on all sides. Set aside while preparing other components.

Cut potatoes into thick wedges, about half an inch each. Toss them in a bowl with olive oil, dried parsley, salt, and pepper until evenly coated. Ensure grill grates are well-oiled or use a grill basket to prevent sticking.

In a small bowl, whisk together mayonnaise and lemon juice until smooth. Add the coleslaw mix and toss gently until every strand is coated. Let sit in the refrigerator while the chicken and potatoes cook. Arrange cucumber slices on the plate just before serving.

Preheat grill to medium-high heat, about 400 degrees Fahrenheit, for 5 minutes. Place seasoned drumsticks skin side down on the grates along with the potato wedges. Grill chicken for 12 to 15 minutes per side, turning halfway through. Grill potatoes for 15 to 20 minutes total, stirring occasionally. Chicken is done when internal temperature reaches 165 degrees Fahrenheit at the thickest part of the thigh.

Remove chicken and potatoes from grill and let rest for a minute. Arrange golden potato wedges, crispy drumsticks, and a generous scoop of fresh slaw on each plate. Add cucumber slices on the side for color and crunch.