The original Parade problem is fairly ambiguous and incomplete. Therefor there is no "correct solution". Other formulations typically include the conditions that the host will always open a door, and that door will always contain a goat as explicit assumptions. In that case the math and logic are pretty straightforward.

But in the setting of an actual repeated game show, I think the more realistic case to keep the show interesting is that the host will mix up his strategy, i.e. sometimes he opens another door, sometimes he doesn't. If it's a long running daily or weekly show, if he always opens a door with a goat, the public will work out pretty quickly that it's advantageous to always switch.