Well, Dibdin's songs came from the stage rather than the parlor, so if you mean "parlor" in this widened sense, I'd suggest:

Rule Brittania (Thomas Arne)
Saylors for My Money (Martin Parker)
Neptune's Raging Fury, or The Gallant Seaman's Lament (Martin Parker, revised by John Phillips)
A Wet Sheet and a Flowing Sea (Alan Cunningham)
Sling the Flowing Bowl (attr. to Richard Sheridan)
Little Billee (William Makepeace Thackeray)
Married to a Mermaid (James Thomson and/or David Mallet, borrowing from Thomas Arne)
The Canadian Boat Song (Thomas Moore)
A Life on the Ocean Wave (Epes Sargent, Henry Russell)
On Board of the Kangaroo (Harry Clifton)
The Bold Fisherman (George W. Hunt)
Strike the Bell, Second Mate (derived from "Ring the Bell, Watchman", by Henry Clay Work)
On Board of the Arethusa (Prince Hoare, Shield)
Jack Was Every Inch a Sailor (music hall)
I am the very model of a modern major general (Gilbert & Sullivan)
The Chivalrous Shark (Wallace Irwin)
Jack Robinson (Thomas Hudson)
Asleep in the Deep (Arthur J. Lamb, Henry W. Petrie)
Down in a Diving Bell
The Wonderful Crocodile, and The Great Sea Snake (both based on the music hall song "The Great Meat Pie")