Here is the situation.

Parasite has 3 virus counters on it and is attached to Data Raven. The runner also has Data Sucker with 1 virus counter.

When the Runner approaches Data Raven can they avoid choosing between taking a tag or ending the run by using Data Sucker to destroy the ice (lower its strength to 0, triggering parasite)?

Does this scenario play out any different if they have Ice Carver?

1 no. Ice is encountered at 3 on the structure of a run and ice breakers interact at 3.1.

2. Yes. Both when encountered effects trigger at 3, but the runner's trigger first and in the order of his choosing, which will destroy the ice via parasites triggered ability.