When the rain starts or when i set the rain manually with commands, I go to the Video Settings and set the particles to Minimal and the sound of the rain just fades out to silence. This happens in any game mode.

Steps to re-create

1: On any game mode set the weather to Rain or Thunder

2: Pause your game and go to Options > Video Settings

3: Find Particles and set it to Minimal

4: Go back to your game and make sure your volume is up to hear the silence of the rain!