I think we have other threads on this topic, but here are a few that I do for Dad's Day (in addition to some that have already been listed):

My Guy
That Silver-Haired Daddy of Mine
Daddy's Hands (Holly Dunn)
Daddy, Don't You Walk So Fast
Don't Jump Off the Roof, Dad (Garrison Keillor's version)
When Father Papered the Parlour
My Heart Belongs to Daddy
Something Wonderful (From The King And I)
Love Without End, Amen (George Strait)
Oh, My Papa
Dance With My Father (sung by Luther Vandross)
Kids! (From 'Bye Bye Birdie')
Grandpa (Tell Me 'bout the Good Old Days) (The Judds)
Grandfather's Clock



PS, Isn't the song from The Fantasticks titled "Plant a Radish"?

There's another song the 2 fathers sing in that musical about their kids: (They Did It 'Cause) We Said No.

Then there's the lullaby that goes
"Hush, little baby, don't say a word.
Papa's gonna buy you a mockingbird ... "