But all it would take, really, is ONE sick parrot in a smuggled load. The Spanish flu epidemic started when one sick woman stepped off a boat here.
I've heard a million stories about where the colonies got started... one of them involved PT Barnum and the parrots escaping. They live in large colonies and make a lot of noise, although they are pretty to look at. I've heard they got their start in Bridgeport, Connecticut in a seaside area called "St. Mary's By the Sea".
Getting back to the main point - mosquito borne diseases... it's not hard to imagine them arriving on aircraft. Suppose a mosquito gets squashed on a windshield of a jet... the eggs inside would survive the six hour flight from Africa and possibly be washed off by the rain into a swampy area. After all, Tiger mosquitoes came over in the residual water in used tires shipped to America for re-treading.
However, if you want to pin this one on Castro and Saddam, hey, why not?