It means "Don't spoil my dream by being pessimistic."
You have a dream about something wonderful that you hope will happen. Someone says "but if X happens, your dream won't happen." Their pessimism has spoiled your dream.
The idiom is an example of a "metaphor." Instead of saying "your dream," it talks about a wonderful parade, planned to honor you. The pessimistic idea is that it might rain on the parade date. The parade might be cancelled. Or maybe nobody will come.
The metaphor could be phrased "I am dreaming of a parade, don't say that it might rain on my parade." This is shortened to "don't rain on my parade," suggesting that the pessimist is actually making it rain.
An example of use:
"I have a good chance of winning," Page said.
"But what if you don't?" her older brother Bryan said.
"You guys just love to rain on my parade, don't you?"
A similar idiom is "to throw cold water" on an idea.
The idiom became the title of a song made famous by Barbra Streisand. Her character, Fanny Brice, is in love with a man named Nick. A friend thinks Nick will break her heart. She is "raining on Fanny's parade." This launches into a wonderful movie sequence. Streisand sings, defiantly, "Nobody, no, nobody, is gonna rain on my parade!" The climax of the song starts at about 3:00 if you won't want to watch the whole scene.
https://youtu.be/-Yfh_CpA9Sk (whole scene)
https://youtu.be/-Yfh_CpA9Sk?t=180 (climax of song)