Too numerous to mention...parting songs are a huge dimension of traditional song, literally hundreds stem from every region of Britain and America, let alone Canada, Australia, not to mention non-Anglo-American traditions. A few favorites that come to mind:

The Dawning of the Morning (Irish)
My Dearest Dear (Appalachian, Sharp)
Ten Thousand Mile (Sandburg/Burl Ives)
Farewell to Tarwathie (Scots)
Swansea Town (Welsh)
Goodbye, Old Paint, I'm a-Leavin' Cheyenne (cowboy)
Goodbye, My Honey, I'm Gone (Grant Bros. and others)

Country music's loaded with them. You could start with this and work forward in time:
You've Been a Friend to Me (Uncle Dave Macon and others)

Traditional and modern blues songs would have to be included, a few at random being:
See That My Grave Is Kept Clean (Blind Lemon Jefferson)
Yellow Dog Blues (Sam Collins)
Lost Lover Blues (Lottie Kimbrough)
Leaving Home Blues (Walter Rhodes)
If You Haven't Any Hay Get On Down the Road (Skip James)

From folk-rock, a couple of notables:
Who'll Buy Your Chickens When I'm Gone / Paul Clayton's Who's Gonna Buy You Ribbons
and the song based on it, Dylan's Don't Think Twice

Clapton's "Layla" is one of the most shattering -- though it may not fit your definition of "parting," that's surely what it's about.

Obviously there are many more, this is just off the top of the head. Bob