Pat the chicken leg quarters dry with paper towels. Sprinkle paprika, garlic powder, salt, and pepper all over both sides, including under any skin flaps. Let sit for a few minutes while you prepare the other components.

Cut potatoes into thick fries about half-inch thick. Heat olive oil in a large skillet over medium-high heat. Add fries in a single layer and cook undisturbed for 4-5 minutes until golden on the bottom. Flip and cook another 4-5 minutes, stirring occasionally until golden and tender. Transfer to a paper towel-lined plate and season with salt and pepper while hot.

In the same skillet with leftover oil, place chicken skin-side down over medium heat. Cook for 12-15 minutes without moving, allowing skin to render and brown. Flip and cook another 8-10 minutes until internal temperature reaches 165 degrees F. In the last minute, brush with BBQ sauce and let it caramelize.

In a small bowl, toss coleslaw mix with mayonnaise, vinegar, salt, and pepper. Adjust mayo or vinegar to your preference, tasting as you go for the right balance of creamy and tangy.

Arrange the BBQ chicken and fries on a plate alongside coleslaw. Garnish chicken with fresh chopped parsley. Serve immediately while everything is hot.