The story told by this song is remarkably consistent between versions, suggesting it is not particularly old. The mother of the singer keeps a boarding house. While the boarders are out, the famed Shallow Brown courts and wins the singer's mother. The father wastes away in sadness... but only because she returns the very next day!
Recordings of this song started showing up in the 1950s (Richard Dyer-Bennet, Burl Ives). The origin seems to be Whitehead & Harris, Six Sea Chanties (1925). Lighter at the Mudcat Café reports:
In the mid 50s, Taylor Harris wrote in the magazine "Sing Out!" that he'd collected "Hullabaloo Belay" from a sailing-ship man who could remember only the first stanza and the chorus. Facing a deadline, Taylor Harris decided to add some additional stanzas of his own invention. Thus the song as it is known today.
The song is distinct from the Scottish hunting-song-turned-lullaby Hilli Ballu or the Robert Burns song Hee Balou, although the spirit is similar.