There are a lot of differences genetically between say an Umbrella Cockatoo and a Blue & Gold Macaw which prevent interbreeding. However, there is a certain amount of hybridyzing in the parrot world. For example, many Macaw species can make hybrids.
For example:
Miligold= Blue & Gold Macaw+ Military Macaw
Catalina Macaw= Blue & Gold Macaw + Scarlet Macaw
And there are a bunch of these. Not my fave (I think the birds end up looking like rainbow sherbet), but they do have some amazing colors.
I do know there are a bunch of ones with Conures too, like a Sunday which is a Sun + a Jenday (or Nanday maybe, I can't remember), and the rescue I got Penelope from has a St. Thomas Conure + Nanday Conure hybrid. I recall reading an article years and years ago too about a lot of Green Cheek/Black Cap/Maroon Bellied Conure hybrids too.
Some of these do happen in the wild, but most are captive stuff. I'm not a huge fan of it, because it's mostly done to make a "prettier" bird, which lends to the "birds as furniture or possession" mindset that we all know leads to abuse and neglect. Still, from what I've heard, certain ones of the Macaw hybrids help dilute nippiness that can happen in certain Macaw species (specifically Scarlet). Still, that may just be the individual birds. I have read stories about hybrids that just accidentally happened because the owners had a bunch of Macaws and two "just fell in love," but personally, if I ever get a bird laying eggs, the eggs are hitting the trash can ASAP, especially if there's a chance of the babies being hybrids.