If you look closely, you can see a picture of Dumbo under the ride's sign. They're pretty blatant about it. The parrots are a nice touch, though.
Also, I'm pretty sure that's the exact same "Flying Elephant" ride they have at Six Flags Great Adventure. Let me find a picture of it...
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Yeah, pretty sure it's the exact same ride, except the ones at Pleasure Beach were given an ugly paint job. They honestly look like they should be part of the film's "Pink Elephants on Parade" sequence.
One more thing - what's that animal on the blue elephant in the center of the ride? A beaver?