The publisher George Thomson commissioned Boswell, among others, to write verses to specified melodies. According to James Cuthbert Hadden (George Thomson: The Friend of Burns : His Life and Correspondence, 1898, 220-221) this air was previously called 'We'll Make a Bed in the Barn'; nobody seemed to be sure whether it was Irish, Scottish or English. Boswell thought the tune vulgar in despite of Haydn's arrangement, but supplied three verses for it: 'as few people sing more than one stanza ... an English, an Irish, and a Scotch stanza, which your vivacious friends may take ad libitum.'

Thomson apparently omitted the 'Scotch' stanza when printing it. Presumably 'The Parson boasts of mild ale' is the first line of the 'English' stanza. The rest I don't find at a cursory look, but it's probably out there somewhere. Hadden's book, which will provide plenty of contextual information (though no more, I think, about this particular song) can be seen in facsimile at the Internet Archive.