Periodically parroted European writer's style (3,4)
I believe the answer is:
art deco
'style' is the definition.
(decorative style)
'periodically parroted european writer's' is the wordplay.
'periodically' indicates alternate letters (letters must be selected 'every so often').
'european writer' becomes 'Eco' (Italian writer Umberto Eco).
The alternating letters of 'parroted' are 'artd'.
'artd'+'eco'='ART DECO'
(Other definitions for art deco that I've seen before include "Architecture and design movement of the 1930s" , "1920s and 30s style of art and architecture" , "Early C 20 style" , "Style popular in the 1920s and '30s" , "Highly geometric style" .)