There are many songs of the kind you mention, but you'll need to be much more specific or people will be recommending things like Cushie Butterfield (much too late, and anyway just a straightforward parody of Polly Perkins with no word-play in it) or irrelevant "Child" ballads (those mentioned so far were unknown on the 19th century popular stage, though there were some parodies of others).

There was a huge corpus of "Cockney", "Irish" and "Jewish" songs, in particular (also many based on the "yokel in the big city" theme, frequently featuring Yorkshiremen), which might work for you; puns, malapropisms and references to other songs all over the place. Where to start? The book I mentioned earlier would provide plenty of ideas, and, no doubt, would most of the others. You might also trawl the Bodleian site ("browse for words in titles"). Check your university library for a set of The Universal Songster, which is of the right period.