So, "How might a fictional parasitic superorganism like the Halo franchise's "Flood" function?" The short answer is only through a conjugation of speculative biology and pseudoscience (often referred to here as 'magic').
The concept of the Flood as set out in the Wikipedia entry link provided by the OP is a turgid morass of pseudobiological nonsense. Real biologists are urged to stay away. It will hurt your brains. Simply put, the Flood couldn't reasonably function as described. At least, not in any way that makes biological sense.
However, the concept can be streamlined into something more feasible. Realistic is something best left behind.
Parasitic organisms typically hi-jack the bodies of their hosts to provide sources of nutrition and a place for their reproductive cycle to commence. Parasites jettison many biological features that are necessary for their survival and propagation. For example, digestive organs are gone. They are unnecessary when your host does all the digesting for you. Nervous systems are simplified.
Essentially parasite is an organism that consists almost exclusively of the means to attach itself to its host's body and remain there, and a hypertrophied mass of reproductive organs. Parasites are simply egg laying machines to ensure the survival of their next generation.
There are some mobile parasites that merrily bore their way through their host's bodies. They are also reproductive engines that absorb nutrients directly from the host's tissues and bodily fluids.
The Flood may seem like a parasite, but in many ways it is acting quite differently. It has similarities to the way viruses attack and take over cells and the fungus that creates zombie ants.
Ophiocordyceps unilateralis is an entomopathogen, or insect-pathogenising fungus, discovered by the British naturalist Alfred Russel Wallace in 1859, and currently found predominantly in tropical forest ecosystems. O. unilateralis, also referred to as a zombie fungus,[better source needed] infects ants of the Camponotini tribe, with the full pathogenesis being characterized by alteration of the behavioral patterns of the infected ant. Infected hosts leave their canopy nests and foraging trails for the forest floor, an area with a temperature and humidity suitable for fungal growth; they then use their mandibles to affix themselves to a major vein on the underside of a leaf, where the host remains until its eventual death. The process leading to mortality takes 4–10 days, and includes a reproductive stage where fruiting bodies grow from the ant's head, rupturing to release the fungus's spores. O. unilateralis is in turn also susceptible to fungal infection itself, an occurrence which can limit its impact on ant populations, which has otherwise been known to devastate ant colonies.
Source: Orphiocordyceps unilateralis
Interestingly, and quite chillingly, this fungus is very specific about which species of ants it turns into zombies. The ant species it doesn't into zombies, simply die. Also, the fungus is able to produce specific chemical cocktails for controlling the brains of its ant hosts.
These two concepts can be used as the basis of a more feasible Flood. The organism responsible could be a virus. It enters cells and takes over their metabolic machinery. By colonizing the neurological systems of sapient organisms they can create zombified hosts. Alternatively, the agent responsible for the Flood could be a highly evolved version of the zombie fungus. However, viruses can survive extreme conditions and especially the extreme conditions of space itself.
In general terms, the virus and zombie fungus concepts provide a feasible biological basis for the Flood. This is still highly speculative biology, real biology itself provides analogues for the biological mechanisms necessary for such a parasitic superorganism.
The next stage is to add pseudoscience to make it full workable (in theory, at least). This is a bit of mental "magic". Assuming the existence of telepathy, it is not entirely unreasonable to assume that a telepathic virus could develop. Once this telepathic virus infects the brains of preferrably sapient organisms, it can take command of its host's knowledge, skills and memories. It can alter and control its host's behaviour to firstly maximize infection of other host individuals, spreading the infection through populations of its current sapient host.
By means of telepathy the growing infection can share the knowledge, skills, and memories it has acquired. This enables it to grow and develop a group mind making it a formidable adversary for its hosts.
Infected hosts may behave in ways as if they continued to act in ways normal to their species. for example, infected humans would behave like normal humans except their behaviour would be modified to ensure the Flood's infection continued to spread until the entire human population was taken over. Presumably the same thing would happen with any sapient populations.
This concept of a Flood-type of parasitic superorganism has been constructed by combining plausible biological models of viral and fungal mechanisms for usurping their hosts in favour of their parasitic organisms, and the pseudoscience of telepathy to enable the infection to develop a group mind. It has streamlined out the overly elaborate persiflage of the original Flood, but as science-fictional explanations go for an alien superorganism it is passable.