Every time my Pokemon is faster than the opponent, but then is Paralyzed, the opponent seems to always go first . . .
Paralysis does cut Speed. It will lower it by 75%. So if my Mew has 100 Speed, and got Paralyzed, it will have 25 Speed.
Yep, it lowers the Speed stat by 75%.
Yes, the Speed stat is dropped. a 75% drop.
SO if you have a Pokemon with 100 Speed, it will then have 25 Speed after Paralysis.
Source: EXperience