In The Penultimate Peril, Olaf tells the Baudelaire children that his parents were killed by poison darts.
Much earlier in the book, Kit Snicket tells the Baudelaires that "during intermission I followed them to the snack bar and slipped them a box of poison darts before Esmé Squalor could catch me."
A while before the Vernacularly Fastened Door was opened, Olaf told the Baudelaires to ask Dewey Denouement about their parents, a particular night to the opera (La Forza del Destino), and a box of poison darts. The details of this are still unclear, but it implies the Baudelaire parents killed Olaf's parents
Whether Lemony or Jacques Snicket knew of their sister taking a box of poison darts to the Opera to allow Beatrice and Bertrand to kill Olaf's parents is unknown; though, given that Lemony loved Beatrice, we can assume he would have interfered if he had known.