If a card forces the corp to derez a card does parasite remain hosted there or go to trash?

This was in the official FAQ; Parasite remains hosted and continues to build counters. However the ICE has no Strength until rezzed, so cannot be trashed through Parasite until it's rezzed again.

so if the corp rezzes the ice and it has strength 0 or less it's immediately trashed?

Yes.

so if the corp rezzes the ice and it has strength 0 or less it's immediately trashed?

Actually it depends on how many virus counters are on Parasite. Even if the ice it was played on is de-rezzed, it will still gain virus tokens at the beginning of each of your turns.

so if the corp rezzes the ice and it has strength 0 or less it's immediately trashed?

Actually it depends on how many virus counters are on Parasite. Even if the ice it was played on is de-rezzed, it will still gain virus tokens at the beginning of each of your turns.

how does it depend? if the ice has 0 or less strength for any reason (ice carver, counters, etc) as soon as it is rezzed, and parasite is installed on it, it is trashed.

how does it depend? if the ice has 0 or less strength for any reason (ice carver, counters, etc) as soon as it is rezzed, and parasite is installed on it, it is trashed.

Depending on how many virus counters, the ice could still be trashed if it has more than base 0 strength.

how does it depend? if the ice has 0 or less strength for any reason (ice carver, counters, etc) as soon as it is rezzed, and parasite is installed on it, it is trashed.

Depending on how many virus counters, the ice could still be trashed if it has more than base 0 strength.

right, but they weren't referring to base strength. they were referring to its current strength when rezzed again with the parasite already on it being at 0 or less. besides, if the ICE had a base strength of 0, it would have been trashed as soon as parasite was installed in the first place. therefore, it is an impossible situation.