No, there's a fundamental difference here. I wouldn't disagree if viruses could reproduce but only inside certain cells - would would be no different than parasites as you say.

That isn't what happens.

When cordyceps infects an ant, it controls the ant, and grows out of its head. Birds eat it, and it's spores end up in the bird's poop. Edit for more detail: cordyceps literally eats the ant's tissue, and uses that biomass to grow and produce the spores that will end up in the bird's poop.

The fungus requires the bird for its life cycle, but the fungus grows and produces spores on its own.

So what happens with viruses is that they cause the cell itself to produce more viruses. Viruses don't reproduce on their own at all regardless of environment.

This is specifically what makes viruses so weird and prompts questions about whether they are even alive, because they are the only organism that doesn't reproduce. They don't lay eggs, have babies, mate, reproduce asexually, eat, or even have a life cycle so to speak.