Kevin Sweeney explained this process perfectly on twitter a few weeks ago.For real. If you need a search firm for this hire, you shouldn't be an AD.
"AD fires coach, hires search firm to help him hire new coach and is now in good graces with said search firm so they recommend him for better AD jobs that they have the contract for. And the cycle continues."
Being that Brad was recommended by Parker Executive Search, and ultimately hired here, I would bet he will hire them to keep that relationship strong if he has any interest in ever becoming a P5 AD.