I forgot the name of the CD, as I gave it to a friend a while ago who had recently had a kid, but, from another point of view, Sugar Hill put out a CD of fathers singing lullabies for their kids with Doc Watson, Tim O'Brien, The Red Clay Ramblers, The Seldom Scene and others in their folk & bluegrass stable of artists.

Papa's On The Housetop - Jim Kweskin (with and w/o the jug band)
My Old Man - Jim Kweskin Jug Band (Yet another song by that name)
The End Of The Rainbow - Richard Thompson
Your Dad Did - John Hiatt
Me And All The Mothers, Dilated To Meet You and any number of other father songs written by Loudon Wainwright III
Tom Paxton has a couple, for that matter.
No Deeper Blue - Townes Van Zandt
Papa Was A Rolling Stone - The Temptations
The Miller's Will - New Lost City Ramblers
Come Home, Father - Joan Morris, William Bolcom & Clifford Jackson (the Henry Clay Work Original) or Poor Little Bennie - Bela Lam & His Greene County Singers (the country version from a '78).
Hang On The Bell, Nellie - Joe Hickerson or The Chad Mitchell Trio
Everybody Works But Father - ?
My Father's Son - Ricky Skaggs
Carpenter of Wood - The Bluegrass Cardinals


Maybe more later...