The plot of Parasyte is essentially a much more violent version of 'Invasion of the Body Snatchers.' Parasitic aliens (we assume, the audience never gets a definitive origin story) covertly arrive on earth and begin taking over human bodies and eating humans indiscriminately. They accomplish this by "eating" the humans' heads and replacing them with their own bodies. However, some humans manage to stop the parasites from taking over their brains. But as a result, the parasites eat and fuse with a different part of the body in order to survive. The Main Character, Izumi, is one of these people. The parasite was only able to take over his right arm.
Because of his unlikely situation, Izumi ends up between a rock and a hard place. Since the only people who discover the parasites' existence are subsequently eaten, Izumi is the only living human who knows of their existence (at least in the beginning). However, Izumi's parasite, known as Migi (right), threatens to kill anyone who he attempts to inform. Nonetheless, the duo eventually grow to be allies as their number of mutual interests increase.
I have only one complaint about this series, regarding its pace. In my opinion, the series "crescendos" too early. The most emotional events of the series take place in episodes 5 and 12 (out of 24: less than halfway through the series), rendering the remaining half of the series a little deadpan and slow-paced. If I had to quantify the difference, I'd say that the first half of the series is a 10 but the second half is a 7. But when compared to other horror series, 'Parasyte' is still rather amazing.
Parasyte is a Sci-Fi/Fantasy/Horror series; but more than that, its very philosophically and emotionally stimulating. This series delves into the issue of the existence/purpose of altruism, which is left rather open-ended. I took the larger meaning to be that altruism, while pointless in-and-of-itself, is an unavoidable side effect of happiness, because comprehending happiness requires empathy. But every viewer will draw their own individual conclusions.