It also depends on the role.
For example, roles in Rides and Attractions (such as Team Leader and Assistant Manager) require you to know how to operate the rides, as well as manage a team. Whilst I think in most cases they will advertise the role both internally and externally, the job would almost always go to someone internally as a promotion, for the simple reason that that person has more experience with the rides themselves, and so creates a smoother transition. That's not to say it's always the case, but it's certainly a massive plus.
Other roles tend to seek out knowledge of the particular park you're applying for, but the level they're thinking of is easily picked up if you've visited a few times. Some roles would prefer theme park experience (even working in retail at a theme park is different to working in retail on the high street), but that's not particularly essential.
Tl;dr - some supervisory roles within operations are preferred for internal candidates as promotions, but this doesn't stop external candidates being considered (and being hired if they're the right fit for the role).