(covers information from several alternate timelines)
Interspatial parasites were organisms from the Sphere-Builders' transdimensional realm.
In October of 2153, Captain Jonathan Archer's hippocampus became infected with these parasites when the starship Enterprise NX-01 encountered a spatial anomaly in the Delphic Expanse. The parasites inhibited certain synaptic functions, causing anterograde amnesia, which prevented him from forming new long-term memories.
As the parasites were from outside normal space-time and existed in a state of interspatial flux, none of Doctor Phlox's treatments had any effect. Archer was relieved of duty while Phlox worked on a cure. A year later, the Xindi were successful in their attempt to destroy Earth. Phlox returned to Denobula to work on a cure while Archer went with the rest of what was left of Humanity to establish a colony on Ceti Alpha V.
Phlox consulted with his world's finest neurosurgeons and quantum physicists, who concluded that there was no way to eliminate the parasites short of vaporizing Archer in a subspace implosion. A decade was spent developing an alternative way of dealing with the parasites.
By 2165, Phlox discovered that he could eliminate the parasites using a highly-focused antiproton beam powered by a starship's warp core. Traveling to Ceti Alpha V, Phlox and T'Pol performed the procedure on a cluster of parasites, and discovered that, due to the parasites existing outside of normal time, by destroying the parasites in the present, they had effectively neutralized them in the past as well. They realized that, if they could eliminate all of the parasites, they could change history in such a way that Archer was never infected in the first place, allowing him to remain in command and possibly saving Earth in the process.
Their work was cut short when Enterprise came under attack by Xindi forces seeking to wipe out the last remnants of Humanity, critically damaging the equipment they were using to eliminate the parasites. With no other option remaining, Archer and T'Pol initiated a subspace implosion, destroying the Enterprise and killing themselves, but also eliminating the parasites in all time periods. A new timeline was thus created in which Archer was never infected, leading to a completely different outcome in Enterprise's mission. (ENT: "Twilight")