Melt the butter with the olive oil in a small skillet over low heat. Add the garlic and cook gently until fragrant but not browned, 1–2 minutes. Remove from heat and let cool 5 minutes.
In a medium bowl, whisk together the mayonnaise, Parmesan cheese, lemon juice, Italian seasoning, crushed red pepper flakes, black pepper, and kosher salt until smooth.
Slowly stream the warm garlic-butter oil into the bowl while whisking until the sauce is emulsified and glossy.
Whisk in the warm water a little at a time until the sauce is thick yet pourable and clings to a spoon in a thin sheet.
Let the sauce rest 10 minutes to meld, then taste and adjust seasoning to preference.
Toss hot, cooked wings with enough sauce to coat, or serve as a dipping sauce. Sprinkle fresh parsley over sauced wings for serving.
Cover and refrigerate leftovers up to 1 week; warm gently and stir before using.