Yes, It's true. It's illegal to own one in California and at least eight other states as well. In California, Monk parakeets (aka Quaker parakeets) were banned in the 1970s, primarily because of destruction they were causing to farms in the Central Valley, where they were congregating to eat cherries, grapes, corn, and pears. The state authorized local law enforcement officers, health officers, and agricultural enforcement officers to confiscate any parakeets seen or reported. Often the job is handled by the local Humane Society.
Since then other states, but especially Florida, have been having serious problems with power outages caused by the birds. The Monk parakeet is the only bird in the parrot/parakeet family that builds nests out of sticks, and their favorite location for them is in power substations and on electrical transformers. Because the nests are communal ones, they're huge! In 2001 alone Monk parakeet nests caused over 1,000 power outages in Florida.