To go one level deeper: It's actually an old Pete Seeger folk song.

So the Byrds were covering a Pete Seeger song, who used lyrics by Ecclesiastes.

It probably is the source of those particular words in Modern English in that particular arrangement. Something very close appears in Wycliffe's Bible and starts like this:

Alle thingis han tyme, and alle thingis vndur sunne passen bi her spaces.

See https://en.wikisource.org/wiki/Bible_%28Wycliffe%29/Ecclesia....