Elrond's Council was a one-time thing, convened out of need and comprised of people who were simply available. It seems from THE HOBBIT that Thranduil was not involved with the White Council. Few of his people would have been Eldarin in origin, and apparently none were of Noldorin origin. The Elven-wise were all or mostly Noldor, it seems, though Cirdan was one of their number and he had never been to Aman.

That is not to underscore Thranduil's importance in Elven politics, however. He was the the most powerful Elven lord in the Third Age. His people were more numerous than the peoples of Lorien, Imladris, and Lindon at the end of the Third Age (after the nearly final exodus of the Noldor) and it is possible (but not really provable) that he governed the majority of Elves in Middle-earth throughut the Third Age. Tolkien does say the Silvan Elves were more numerous than the Eldar.

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Parma Endorion: Essays on Middle-earth, Revised Edition