Is H.G. Wells' The Wonderful Visit, though not strictly a science fiction novel, the first such novel to use the parallel universe theory? Consider this dialogue, spoken by an angel-like creature who enters our world and by the vicar who promptly shot him:
"It almost makes one think that in some odd way there must be two worlds as it were...."
"At least Two," said the Vicar.
"Lying somewhere close together, and yet scarcely suspecting...."
"As near as page to page of a book."
"...there may be any number of three dimensional universes packed side by side, and all dimly dreaming of one another. There may be world upon world, universe upon universe. It's perfectly possible. There's nothing so incredible as the absolutely possible."
The Wonderful Visit, Wells' second novel, is in fact several things: a fantasy, a drama, a comedy and a satire. It's even a love story of sorts. It's easy to imagine it adapted as a British TV series and it was adapted as a film (La Merveilleuse Visite, 1973, by Marcel Carné) and an opera (La Visita Meravigliosa, by Nino Rota, in 1969).
Inspired by John Ruskin's remark that an angel appearing on earth in Victorian England would be shot on sight, Wells guides our innocent angel through needling, sarcastic, patronizing hell, though that hell is a small village in southern England filled with gossips and busy bodies.
The height of the book is a deliriously wry and funny scene wherein our angel makes his debut as a violinist in "polite society". I wouldn't be surprised if the sharp writing here was an influence on P.G. Wodehouse. Reading The Wonderful Visit, it's difficult to keep in mind it was written in 1895.