Atheism doesn’t generally have institutions.
Are you really specifically meaning atheism, as in, to quote Merriam-Webster “a lack of belief or a strong disbelief in the existence of a god or any gods” ?
Because, if you are, then it’s not very well framed to ask what it’s parallel is, because it’s not that kind of a thing.
Perhaps you mean, what is a non-Christian form of what Christians call (and sometimes own as a Christian-specific thing) stewardship?
That would make much more sense to me and seem more answerable.
In which case I would tend to say: environmentalism. You could also still use the word stewardship, I would say. Perhaps “human stewardship of the earth.”