The Collaborative International Dictionary of English v.0.48:

Conure \Co*nure"\ (k[-o]*n[=u]r"), n. [NL. conurus, fr. Gr.
   kw^nos a cone + o'yra` tail. The name alludes to the tapering
   tail.] (Zool.)
   An American parrakeet of the genus Conurus. Many species
   are known. See Parrakeet.
   [1913 Webster]