When we think of the title of the sister poem to "Parting at Morning" ("Meeting at Night") we get that the similarities between the two are no accident. They're supposed to look opposite but also equal to one another because they run parallel in terms of themes and context.
The "at" in both titles acts a bit like the union between the two opposites: parting/meeting, morning/night. So just like everything else in this super succinct poem, the title itself is meant to be poignant and to the point. Each title tells us exactly what the each poem is about.
In addition, "Parting at Morning" is what's called an aubade (a song or poem about lovers parting at morning). So we get the feeling that Browning was probably looking to keep things as simple as possible by using a title that literally defines what the poem is about. And since aubades weren't all that common in the 19th century, it's not like he had a whole lot of competition in that regard. Why not keep it simple?
Besides the technical stuff, we also recognize that the connotations of "parting at morning" also bring to mind lovers, unions, a passionate night before, etc. And yet, this poem is about parting—it's about what happens when lovers leave each other. By the time the speaker gets to "Parting at Morning," he's ready for that "world of men" (he's done with the romantic isolation of the night before) and the path he needs to walk without his lover. So the title prepares us for his separation from his lover and the idea that it's time to get down to business—only this time it's worldly business and not the sexy kind.