Player A has a Shadow Parasite in play. Player B has a STR 1 character. Player A triggers an effect that gives it -2 STR, then triggers the parasite's response. How much STR does the parasite gain?

It would gain 2 STR I would think. Even though the other character only had 1 STR to lose, the STR modifier was still -2 and Shadow Parasite reads:

Response: After a character's STR is reduced by X, Shadow Parasite gets +X STR until the end of the phase.

I'm sure ktom will have an elegant example ready to explain how exactly this works, but I personally feel like it's actually the opposite way. The question is what the phrase "After a characters STR is reduced by X" means. Is X the total STR reduction of the effect applied to the character (as Skowza argues) or is it the net change in STR the character experiences? I would say its the latter, the character has gone from 1 to 0 STR, making X only 1.

Ok, alpha. Let's say that 1 STR character gets the -2. It's effective STR is 0 instead of -1, but how much was its STR reduced by? Let's say that right after the -2 is applied, I throw another +1 at it. What is the STR now? Still 0 because the whole -2 effect is there to be dealt with, right?

So the amount the character's STR was reduced by is the value of the -X modifier applied, not the net difference in the effective STR used after the actual lowering happens/is applied. If the character's STR was only reduced by 1, why do you need +2 to bring it back to normal?

Skowza had it right.

Curses! Foiled again!

Thanks for the answer.