Mr. Poe claims the fire killed both the Baudelaire parents (Bertrand and Beatrice), although there is no mention of their bodies being found, leaving the Baudelaire children to later suspect one of them may be alive.
The three Baudelaire children were not home at the time and thus escaped the fire. They were encouraged by their parents to go to Briny Beach that day. If the Baudelaire parents were aware that the fire would occur, they were unable - or chose not - to save themselves, despite the mansion having a secret passageway to 667 Dark Avenue.
Page 13 of the Snicket File state that "experts now suspect that there may, in fact, be one survivor of the fire". However, it is not stated which "fire" the page refers to. The Baudelaire children initially believed that the file referred to the fire that consumed their house because the page also had a photo of their parents and had been filed under "Baudelaire". Quigley Quagmire believes that the sentence is referring to his survival of the fire which destroyed his own home and killed his parents. Neither theory is ever confirmed.
It is unclear if Kit Snicket was aware of the Snicket File, but she believed the Baudelaire parents to be dead since she refers to the Baudelaire children "losing their parents." The Bad Beginning: Rare Edition makes reference to a survivor of a fire hiding within the City's Fountain of Victorious Finance, but it does not confirm the individual's identity.
The precise cause of the fire has never been established and no one was ever charged for setting it. The mystery of the fire has been a large debate in the series fandom for a long time. One must also remember that there are three continuities (book, movie and TV series) so the reason for the fire may differ in all of them.
One major theory behind the fire is Count Olaf. Olaf has a history of starting similar fires and admits to being guilty of "arson." His possible motives include getting the Baudelaire fortune and taking revenge on Beatrice and Bertrand Baudelaire for murdering his own parents a few years earlier. However, in The End, Olaf seemed to deny his involvement; when Klaus accused him of starting the fire, he only answered, "Is that what you think?" after a long pause, but it is still possible that he did do it and was only asking if they were sure. In the film, Olaf's tower has a magnifying glass which can set fire from the distance and has a view of the Baudelaire Mansion, although this only applies to the film canon. In the TV series, while Lemony Snicket is narrating inside the Baudelaire mansion, a thin beam of light was being shone in from the outside, igniting the whole mansion on fire, meaning it is arson in the TV series canon.
It is also possible that another member of the fire-starting side of V.F.D. is responsible. In the TV series, a fashionable woman speculated to be Esmé Squalor, is seen setting the Quagmire Mansion on fire. Esmé held a grudge against Beatrice, saying, "I want to steal from you the way Beatrice stole from me." This has led fans to theorize that Esmé is responsible, but that was seemingly debunked when later in the series Gustav Sebald was showing disguises and one of them was the same outfit worn by who burned down the Quagmire Mansion, and later in the show the Man with a Beard but No Hair and the Woman with Hair but No Beard said that they tried to keep the Quagmire parents at Peru, hinting that they set the fire instead of Esmé. However, it is possible the fire may be Esmé in the book canon.
Another possibility is that the fire was a pure accident instead of arson, as the Baudelaires are shown to have extremely bad luck. This may be the cause in the book canon only, as the movie and TV series canons imply arson. For example, the fire may have been caused by one of Violet's inventions malfunctioning. In the book The Bad Beginning, Violet mentions she is inept at cooking and sometimes burns toast.