Green’s Dictionary of Slang

smother a parrot v.

also strangle a parrot
[translation of Fr. argot asphyxier un perroquet, to drink a glass of absinthe]

to drink off a glass of absinthe in a single gulp.

[Aus]Truth (Brisbane) 11 Apr. 9/3: ‘Strangling a parrot does not mean bird-murder. It is Parisian slang: for drinking absinthe.
[UK]J. Ware Passing Eng. of the Victorian Era 227/2: Smothering a parrot (French) Draining a glass of absinthe neat. Derived from the green colour of the parrot.
[Aus]Bulletin (Sydney) 31 Dec. 14/1: In fact, no kind of liquor on this planet has such a wide range of slang equivalents [as absinthe]. The Parisian absinthe drinker usually talks of ‘taking a blue,’ ‘killing a worm,’ or ‘strangling a parrot.’.