My father was born just the year before yours and had exactly the same habit of singing snatches of song, anything from light opera, traditional or the poular songs of his day. I remember him singing what I think is the song you refer to:

My pa give me a nickel to buy a pickle
I didn't but a pickle, I bought some chewing gum
Chew, chew, chew, chew, chewing gum, how I love chewing gum
I didn't buy a pickle, I bought some chewing gum

My pa give me a dollar to buy a collar, etc.

Don't know anything about its origin but it has the feel of the 20s or 30s popular songs to me.

wr.