The wages are what's relevant with Baumol. It's possible rising wages of parents helped generate the administrative burden but 90% of the time it was the flood of cheap student debt via gov, every kid thinking they need to go to college, the massive boom in demand via international students, etc.

The fact university administrators responded to boom in demand/capital by hiring way more administrators is probably not directly a consequence of Baumol either. But it is why the costs of these administrators is so expensive, t which is reflected in tution costs.

Maybe you could blame it because being a uni admin is an easy job to get with little actual output demanded of you, but you still get paid like a high productivity sector employee, so people flocked to it. And if they were paid a wage connected to their individual output they wouldn't be nearly as many of them and tuition costs could go down.

But the universities could also just fire 2/3rds of them and still function while paying the rest normal high wages.