The Blue Parrot Café was an establishment on Sarona VIII, across a city square from the Zanza Men's Dance Palace. It served a blue concoction according to William T. Riker.
When the bridge crew of USS Enterprise-D planned to spend part of their shore leave there in 2364, Jean-Luc Picard told Deanna Troi she ws buying. (TNG: "We'll Always Have Paris")
The name is the same as Signor Ferrari's bar in Casablanca, apparently as an homage to the film from which the episode's title was also taken.