Whisk Cajun seasoning, olive oil, hot sauce, and salt in a bowl.
Coat drumsticks thoroughly with the marinade on a tray.
Let rest at room temperature 20 minutes.
Preheat grill to medium-high (about 400°F).
Grill drumsticks 25–30 minutes, turning every 8 minutes until 165°F.
Melt butter and whisk in garlic, parsley, and lemon juice.
Rest cooked drumsticks 5 minutes, then spoon herb butter over top.
Serve immediately while hot and glossy.
Tip: Keep a cooler zone on the grill to move pieces if they start to burn.
Tip: Do not baste with leftover marinade after the chicken has partially cooked.