Observation: Parrots are not native to the USA, at all. They're not even native to Mexico. These guys should be down in South America, yet I found a flock of them, in Austin. A woman at the park said these were pet birds set free. But, pet birds have had their wings clipped, so I don't think they would have survived the Austin pet (dog & cat) population, or the wild-animals, like coyotes, raccoons, etc..., Besides, these birds flew, beautifully!
LEO Says: Thank you for your observation Ann! They look perhaps like monk parakeets (Psittacara holochlorus). We recommend iNaturalist for help with species identification. There are also some good parrot focused web resources including The Parrot Trust: According to Wild Earth Guardians, "Once, two parrot species inhabited the United States. Tragically, the Carolina parakeet is now extinct. The sole surviving North American parrot species, the thick-billed parrot, is hanging on south of the border in Mexico, and we have a chance to bring it back to its historic range in Arizona and New Mexico. The last confirmed sighting of a naturally occurring flock in the United States was in 1938." On Monk Parakeets, a map prepared by NatureServe shows the range in September of 2007 (see images), not extending (yet) into Texas. However your observation and others (see Texas Monthly article: We got the Monk Parakeet, indicates that the green parakeets are establishing themselves in the Austin area.
There is a Facebook page for naturalized monk parakeets in the Austin area. See The Monk Parakeets of Austin, Texas. Monk Parakeet are not only expanding their range into North America but also in Europe. See ScienceDaily article: The Monk Parakeet: Tracking an Invasive Bird