Since you responded and I had some time to look, I found three sheet music songs. I knew I had two for sure and seemed to remember another but only mentioned two above. For some reason I do not remember, one had to come in two parts, top and lower.
The Parcel Post March--

The Parcel Post March and Two Step--

My Parcel Post Man--


This was published by what began as the Ted Snyder Company in 1908 evolved into Waterson, Berlin & Snyder, Inc., in 1917 when its founder, Ted Snyder (1881–1965), took on two partners – Henry Waterson (1873–1933) and Irving Berlin (1888–1989). Berlin had been Ted Snyder's staff lyricist since 1909.
Yes that Irving Berlin.