For Disney, it's the Magic Music Days program... and as far as I know, it's still going strong. Bands and choirs are welcome to perform. They also host separate events for dancers with the "Dance The Magic" program that also is a part of the Magic Music Days, though it's run by an independent vendor--but it's not usual for them to have 400-600 dancers in a single event.
Magic Music Days events are often done on the weekdays, so the parks can boost their attendance and spending (because performers and watching friends and family all have to pay a park admission) on what are usually slower days that offer fewer in-house entertainment, making it a bit of a win-win for everyone. Performers get the experience of performing at Disneyland, and guests in the park get an extra show.
Most of the theme parks offer opportunities for local groups to perform in some capacity. One friend of mine had her group perform on the old stage in the Fiesta Village area at Knott's. I myself had a group perform at USH on the old Blues Brothers stage as part of a similar program (nowadays, I believe the performances are usually on the Five Towers stage). And SFMM (as well as Knott's and occasionally even Sea World) often host dance and cheer events.
And in a bit of personal trivia, I performed in Disneyland Christmas parades for 3 years in the early 90s, back when it was still the Very Merry Christmas Parade. If I'm being honest, the experience really shaped me as a person, and I've never forgotten it. Really brought me out of my shell and gave me a lot more confidence.