Aubade with ABBA
As fitting as an aubade with ABBA (the band) would be, that's not what we're talking about here. But what's an aubade? Simply put, an aubade is any song or poem that's about lovers parting at morning. They're not so common nowadays so that's probably why you haven't heard the term before.
"Parting at Morning" is a simple kind of aubade with four lines and with a rhyme scheme that fits that ABBA pattern. "Sea" (1) rhymes perfectly with "me" (4) while "rim" (2) rhymes perfectly with "him" (3). So it's not just the poem's succinct word choice that's keeping everything together here. It's also the musical sounding rhyme scheme that makes the poem move in a well-balanced and rhythmic way, like an aubade ought to.
Repeat After Browning
There's also quite a bit of parallelism in the speaker's repetition of "and" in lines 2-4. That helps to keep that rhythm alive and makes the poem look mighty structured and balanced. The balance accents the speaker's awareness of the fact that nature is doing the same sort of thing that he is. The sun has a "path of gold" and the speaker is headed on a path toward a "world of men." So even if the speaker is only working with a few words here, his choices of syntax and diction emphasize the meaning behind the words.
What Meter?
Even if we can't pin down an exact meter in the poem, we can spot a few instances of some similar sounds. For instance, lines 3 and 4 not only look similar in the speaker's repetition of "and" but they sound similar, too. Why? Because of that little anapest sitting right there in the middle. Check it out:
was a path
of a world (3-4).
Hear that unstressed-unstressed-stressed syllabic pattern right in the middle of each line? Every line also ends in a stressed syllable that gives it an extra kick and keeps the poem sounding kind of uniform, even if we don't have a prescribed meter.