I am not a language expert, but I feel sometimes that I can associate "sound alike" words that show the development(or bastardization)of languages.
An example of this is that many people who arrived in the United States from Europe had their names inadvertantly changed because the interviewing immigration officers simply wrote down the name they "heard".
The phrase "Inky Pinky Parlez-vous" or "Inky Dinky Parlez-vous" could have developed because of misinterpretation. Variations in spelling are explained by the fact that many letters of the alphabet sound alike to the listener. Additionally, I doubt that the name of Chicago politician "Hinky-Dink" McKenna had any influence on the lyrics of this song.
The typical Mademoiselle from Armentières song is constructed of a series of "Parlez-vous" after each stanza, then "Inky Dinky Parlez-vous".
It is "understand", "understand", "understand", followed by "indicate you understand", the "indicate" being a mispronunciation of the French word "indiquez" ("Indicate" in English).
Why not consider this interpretation of the meaning of "Inky Dinky Parlez-vous" ? I haven't seen any other good explanation of it's origin.