The Sinestro Corps is a group of fiction character, a villainous analogue to the Green Lantern Corps in the DC Universe. It is led by the supervillain Sinestro.
The Parallax entity is a space parasite that is the embodiment of fear that was imprisoned within the Central Power Battery on Oa. As time passed, the entity became known as the yellow impurity, the cause for the power rings' weakness to the color yellow.
Sinestro, at the time the universe's greatest Green Lantern, was sent to Earth by Supernova in a plot to erase Guy Gardner from history. Booster Gold was assigned to prevent this from happening. To do so, he convinced Sinestro to leave Earth, claiming that he was an admirer from the future, and that his yellow Legion of Super-Heroes flight ring was a tribute to Sinestro. When asked what Corps he belongs to, Booster ad-libbed "The...Sinestro Corps," leading Sinestro to twirl his mustache in thought while mumbling "Of course...Of course."
After Sinestro went rogue, he was banished by the Guardians of the Universe to Qward in the Antimatter universe. When he returned, he wielded a power ring which used yellow energy. After various encounters with Earth's Green Lantern Hal Jordan, he was imprisoned within the Central Power Battery. There he was able to use his ring, which uses fear, as opposed to willpower, as a power source, to awaken Parallax from hibernation. From here, Parallax and Sinestro were able to influence the fall of Hal Jordan and instigate the fall of the Green Lantern Corps, leaving one last Green Lantern: Kyle Rayner.
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Official Site: DC Comics
Links: Wikipedia-Sinestro Corps. DC Database DC Comics-Sinestro Corps.