Lily the Pink is a degenerate form of an American ode to Lydia Pinkham:
a lady who concocted a tonic (largely alcoholic) for treating "women's complaints", and advertised it in magazine ads featuring a portrait of Ms. Pinkham.

The Ballad of Lydia Pinkham

    Let us sing of Lydia Pinkham
    The benefactress of the human race.
    She invented a vegetable compound,
    And now all papers print her face.

    Mrs. Jones she had no children,
    And she loved them very dear.
    So she took three bottles of Pinkham's
    Now she has twins every year.

    Peter Whelan, he was sad
    Because he only had one nut
    Till he took some of Lydia's compound
    Now they grow in clusters 'round his butt.

etc.etc. ad infinitum