Other parks (Teton, Olympic) have allowed sportsmen to participate in the culling in the past but it's usually a lot of effort for the park. I assume primarily they do it for optics & relationships with hunting population than because it's a cost-effective way to solve their overpopulation problems.
From what I've read you get brought on as an 'unpaid contractor' to be a temporary employee and must submit an application and pass some practical shooting tests to participate. I believe exo mountain gear guys published a podcast interview with a hunter that did the Teton one.
I presume they jump through those hoops due to federal law rather than park policy but I could be wrong