Here's my two cents on how the infection/parasite could work:
I’m gonna assume that the parasite was made in a lab from scientific experimentations that have gone wrong (how original, I know). There is two ways to get infected by the parasite, either by getting the parasite injected into your body or through sexual intercourse with an infected individual. The second way being the natural way the parasite propagate itself.
Zombie, Female Zombie and Fat Zombie:
The zombification is basically the first stage of the mutation process provoked by the parasite. At some point, if the subject is strong enough, the parasite will attempt to mutate its host body further into something more suitable to propagate itself. If the host body doesn’t respond adequately to the mutation, the host and the parasite will just die.
Fly:
The giant flies are the result of a lab experimental tests on insects, so it would be a different variant from the parasite that impact humans. In the game, there is an inconsistency on how they reproduce. On one hand they can hatch from giant eggs which implies that they can laid eggs, but in the other hand they can impregnant human female and birth from them. It's redundant and doesn't make a lot of sense in my opinion. However, I was thinking of an alternative that could make more sense. Instead of directly giving birth to a giant fly, impregnated female should give birth to a larva that will then need to cocoon itself before hatching as a flying adult. That way the reproduction cycle would make much more sense in my opinion.
Facehugger Humanoid (I’m not sure how to call it, so let’s go with this name):
This is a common mutation that happen after the parasite try to mutate a male host body further. The fact that it can launch facehuggers at the player doesn’t necessarily mean the facehugger humanoid is the source of them. The way I see it, as facehuggers are quite fragile, it could instead serve as vessel to cary multiple facehuggers (like the poison zombie in Half-Life 2 for example).
Facehuggers:
They are a more efficient form for the parasite to quickly propagate itself, I guess you could call them a natural evolution of their form. They originally are the offsprings of a female host which mutated into a brood mother. Although they don’t get the player pregnant in-game, I’m assuming they actually can to explain how they can reproduce quickly.
Tentacles Boss:
The tentacle humanoid boss is one the lab experimental test conducted on human subjects. Basically, a patient zero that escaped the lab and started the whole parasite infection.
On a side note, taking in count the decayed environment in the intro CGs and in the in-game background, I’d be tempted to say that the infection happened for a long time and isn’t something new. It’s possible that the player was inside an area that was spared so far, until somebody infected propagated the parasite inside that area.