J.R.R. Tolkien identifies Glaurung as the father of the dragons in The Silmarillion.
Glaurung is a powerful dragon created by Morgoth during the First Age. He is described as the first of the Urulóki, or fire-drakes, and is associated with the wars of Beleriand. His major appearance occurs in the tragic story of Túrin Turambar, whom Glaurung deceives and manipulates through his speech and gaze.
Glaurung is not the same dragon as Smaug, who appears much later in The Hobbit. Ancalagon the Black is another First Age dragon, described as the greatest of the winged dragons, while Scatha is a later dragon connected with the Grey Mountains.
The phrase “father of dragons” refers to Glaurung’s place in Tolkien’s chronology and mythology, not to a family tree presented like a biological genealogy. Tolkien’s dragons are creatures bred or made by Morgoth as weapons of war.