The Blue Parrot Café was a club on Sarona VIII, across the 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 was buying. (TNG: "We'll Always Have Paris")
This café was only mentioned in dialogue.