Main issue is IP infringement. However you slice it, you're doing a theme park show with another company's characters. This was a huge issue in 2006 when B&Ts Jack Sparrow was basically just a better version of the Disney version. It's a more minor thing, but even music licensing is a challenge when doing a show--you may have heard the law is any band can perform any song live, which is true, except in a theatrical production. I know a second Moana parody was cut last year, I assume for fears stemming from that.
As for the real people parodied, still have to worry about defamation--tough to prove in this day and age, but not impossible--as well as bad press. Think the blow-up over the Mitt Romney sketch in 2012. One joke in poor taste about Bill Cosby or Trump could get the company lit up on social media and TV.
Partly it's the hypersensitive era we seem to live in. But a lot of it is also HHN is an international event now, not a goofy after-hours thing put on for a couple thousand hard-core local fans. You can get away with a lot more when no one's paying attention to you.