The Parliament of Owls was an organization of talking owls that met during the night to discuss the state of affairs in the land of Narnia. They were not a shadow organization, intending to operate outside the lawful rule of Narnia; they simply met during the night because that was the natural time for owls to be awake. The owls often lamented the fact that other Narnian creatures met during such unnatural times, such as during the day.
The name comes from the literary name for a group of owls: a "parliament." A parliament is also the name of a governing body or council, based on the French word parler which means "to speak." The combination of the two meanings is a pun on the term.
The fourth chapter of The Silver Chair is called "A Parliament of Owls" and features this group extensively. They meet in a dark tower late in the evening to discuss the disappearance of Prince Rilian and how to aid Eustace Scrubb and Jill Pole in the search to find him. They plan to take the children to Puddleglum, the Marsh-wiggle, who would act as their guide through the Northern Lands.