It might make you tired looking at the word, "duty," but in "Parting at Morning," there's no denying that our speaker has places to go and things to do in that "world of men." He's got a duty to himself to not just be a lover, like he was in "Meeting at Night," but to also be his own independent man. No matter how you cut it, we understand that love in the poem requires not just the lovey-dovey stuff, but also the independent dutiful stuff that allows a person to stand on his own two feet.
Duty is everywhere in "Parting at Morning," whether the speaker is referring to himself or the natural world.
Everyone, including nature, seems to have his own path in Browning's poem. And each path symbolizes a person's duty to himself.
We definitely can't ignore all of the personification in "Parting at Morning." And all that personification makes it seem as if man and nature are one and the same. The sun is personified as a man projecting a path of gold and peeking over the mountains, while the speaker prepares to embark on his own path. Even the mountains feel oddly reminiscent of the world of men. There's no doubting then that all of the vivid imagery in the poem is meant to bridge those gaps between man and the natural world.
Man is part of the natural world, but in "Parting at Morning," he's so much so that even nature appears to look a bit like a guy.
The natural world has its own purpose and path just like the speaker who's reminded of that "world of men" which he is inextricably a part of.