The exact word sequence "Don't Rain on My Parade" probably dates to the song copyrighted in 1963 and popularized in the musical and movie "Funny Girl," but the phrase "rain on (someone's) parade" was commonly understood to mean "spoil (someone's) time" or "disappoint (someone)" as far back as the 1910s and 1920s.
The September 26, 1912, Schenectady Gazette, page 5 contains a short piece called "Sprightly Adventures of Mr. Homesweet Home" about some men on an all-male hunting/camping trip. The piece contains the following sentence:
Warmbody says he knew dog-gon well, some gosh-blamed leftover-from-the-summer hen [ed. -- woman] would show up to "rain on the parade," and Mr. Home warned the entire party against letting Horace loose on the veranda of the Pelican Bay House when he first caught sight of the imprints of a pair of French heels [ed. -- women's shoes] on the path near the boat landing...
The June 21, 1917, Narberth (Pennsylvania) Our Town, page 2 contains an ad for Howard's drug store with the following copy:
Far be it from us, oh, patriotic plowmen, to shed rain on the parade, but neither would we have your valorous efforts all in vain! And that is why we are well prepared with all the things which make short-shrift of garden pests and parasites...
The May 30, 1924, Monroe (Louisiana) News-Star page 8 contains a short fiction piece called "The Tale of the Royal Fisherman" with the following exchange:
"This is the place!" said King Bozo, drawing his trusty fishing rod and preparing to cast a bait into the rippling waters.
"Sire," interrupted the Prime Minister, "far be it from me to rain on the parade, but take a slant at yonder signboard!"
Nailed to the trunk of a whoofis tree was a crude signboard bearing the word: "NO FISHING EERE!"
The January 31, 1927, Albany Evening News page 14 contains a short fiction piece called "The Hotel Stenographer" in which a girl is talking about Mussolini:
"No, Kelly," the girl spoke patiently. "I am talking about a deuce who is acting like a king over in Italy. It's going to rain on his parade. ... Bachelors are a mighty happy people, Kelly, and old Mussolini must keep them on his side if he wants to prosper."