Here are two Pokemon Moves that can Paralyze Pokemon: Shock Wave and sometimes Body Slam.
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No, you can't. If you want to paralyze a Pokemon, it has to be awake.
shockwave, stunspore, and most electrical moves in later games have a chance to paralyze targets aswell. some electrical Pokemon also have an innate ability to paralyze like mareep and its evolution line, pikachu etc. its an extremely effective tool to catch legendaries, I've caught palkia in pearl first ball(ultra), suicune 3rd ball (ultra) and raquaza with a pokeball and most other legendaries with relative ease while they were paralyzed and have a chip or red hp
put them to sleep, or paralyze them
Putting a Pokemon to sleep will give you a better chance of catching it.
Click on your Pokemon and you scroll down to change the moves to what you want.