To 'send up' means to parody, or pastiche, or in other ways imitate someone's style in an exaggerated manner for comic effect — as one might say that much of John Gay's 'The Beggar's Opera' [although the phrase would not have been used in the 1720s, being a mid-20c usage, I suspect] was a send-up of the conventions of Handel Opera or Italian opera as sung at the time. Or that many early Disney cartoons were send-ups of musical conventions of the time. Or that many of the plots of Popeye, Bugs·Bunny&Daffy·Duck, &c [& also Charlie Chaplin] were send-ups of Victorian melodrama. So, if Robert Service had written Eskimo Nell in an exaggerated form of his usual style, one would describe him as 'sending himself up'.