Horace and James Smith - Rejected Addresses (1812)
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Archive.org direct link to American edition
Probably the first book of parodies to become a popular success. The inspiration was the re-opening after a fire of Drury Lane Theatre when a monetary award was offered for a ceremonial address. Numerous submissions were turned down so the Smith brothers came up with the idea of writing other rejected addresses done in the style of famous poets such as Byron, Scott, Coleridge, Wordsworth, Southey and Moore. It was a huge success, going through numerous editions very quickly.
(The genuine address was written by Byron as a last resort and not without some infighting that apparently diluted his work. Details start at p166 of Life, Letters, and Journals of Lord Byron - "You will think there is no end to my villanous [sic] emendations." and "You will find a sort of clap-trap laudatory couplet inserted for the quiet of the Committee.")
The linked edition includes a preface from two decades later and identifies the poets being parodied. Even if you don't recognize the targeted poets the Smiths' poems still have good-natured humor:
Bloom, Theatre, bloom, in the roseate blushes
Of beauty illumed by a love-breathing smile!
And flourish, ye pillars, as green as the rushes
That pillow the nymphs of the Emerald Isle!
The "American" edition includes solo work by each brother including "Address to a Mummy", "Lachrymose Writers", "To a Log of Wood upon the Fire" and "Diamond Cut Diamond". There's also a biographical memoir (their phrase) that covers their lives with the same wit.
For the curious, or more appropriately connoisseurs of bad verse, a collection of some of the Genuine Rejected Addresses was also published in 1812. "Before a British Audience I appear -- / Then whence this feeling of unfounded fear?" runs one example. (And if nothing else you may want to check out E.N. Bellchambers' submission starting on p68 - too long to quote but it invokes Bacchus and Melpomene (the Tragic Muse), the invasion of the Goths, Liberty and Melpomene's offspring being Shakespeare, includes long footnotes in French, tosses in Aeschylus and Britannia, and perhaps inevitably brings up the Phoenix at the end.)
A full biography of the brothers was published in 1899 by Arthur H. Beavan.