Here is the situation:

Corp has remote server protected with 1 unrezzed ice. I have Ice Carver and Wyrm installed.

On my last click I install Parasite on that ice.

Beginning of my next turn, Parasite gets a counter.

I run on that remote server, Corp rezzes ice: Neural Katana-strength:3

Is the following statement correct? --> I can pay 1 credit for Wyrm's ability to lower the ice strength, and because of Ice Carver and Parasite with 1 counter, Neural Katana is now strength=0 and therefore Parasite's ability kicks in and Neural Katana is trashed?

If that IS correct, what an incredible way to destroy ice without ever having to break subroutines! Of course, I will probably only get away with that one time, and then the Corp will get wise and not leave Parasite's hanging around.

Yes the ice would be destroyed as it is at 0 strength.

Your logic is sound - it doesn't matter how the strength is reduced to zero, it still triggers Parasite. Ice trashing is a big Anarch strategy, especially when bringing Forged Activation Orders from Criminals.

However, Parasite can only be installed on a rezzed piece of ice.

Furthermore, if the ice is later derezzed, Parasite won't trigger until it's rezzed again because unrezzed (and thus inactive) ice doesn't have strength.

According to the FAQ, Wyrm can only lower the strength of a piece of ice once its strength is equal to or higher than the ice's strength (see also the rulebook p. 11, "An icebreaker’s strength must be equal to or greater than the ice it is interacting with").

So, as noted above, the ice would need to already be rezzed before the Parasite can be put on it.  Then, when you encounter it, if the Parasite already has one virus counter and if your Ice Carver is out, you would need to pay 1 [credit] to raise your Wyrm to strength 1, then pay 1 [credit] to lower the ice strength to 0, then it gets trashed.

Even with all that, yes, it's powerful. :)

Thank you all for the responses.