What makes Lemony so confident about the parents' deaths?

[Optimism is my Phil-osophy]

Which part of book 13 does Lemony say that about? I remember saying that they never saw their father again, but not their parents. Either way, Lemony may be lying to protect Beatrice, or he may be wrong. Evidence of one's death may be forged if the person concerned wishes to fake his own death. See Aunt Josefine's example. She left a suicide letter. Besides, Olaf himself said that she didn't really need to die, just that people thought she was dead. Count Olaf himself took advantage of Jacques' death and the news of TDP to fake his own death. Also, you need to consider what Lemony says in chapter 14, about reports of one's death being often false stories. So the theme of false deaths is present from ASOUE Book 3 to Book 13. Lemony may have purposely or not contributed enough to spread Beatrice's death, even though this death is not real. And besides, remember the time factor. It is not yet clear when Beatrice died. She may have died in R's mansion fire because R calls Beatrice flammable. But the fire occurred at least 15 years after Olaf's death. By this time Lemony had already published TBB, TRR and TWW. Did Lemony know about Beatrice's survival? Maybe not. Maybe he just found out that Beatrice was alive at prom time. Or maybe Beatrice really survived that fire too, but Lemony only discovered it at the time he published TCC.
Other evidence about faking deaths is found on the TBL poster. Clearly the scene showing Klaus, Violet and Sunny's goals was left there to be found and people thought they were dead. They faked their own death. They even tied Sunny's cooking object when they realized it would roll and sink. Booklet 13 Secrets indicated that there would be some clues in that poster, and this is evidently one of those clues.

[Foxy]

I think TE said the kids finally accepted that their parents had died, which is quite a fitting way to end a series, because acceptance is the final stage of grief.

I think there are a lot of theories about whether the parents survived or not, but there's no proof of much of anything.

[Dante]

Sept 8, 2019 8:42:37 GMT -5 Optimism is my Phil-osophy said:

Booklet 13 Secrets indicated that there would be some clues in that poster, and this is evidently one of those clues.

You need to remember that 13SS came out before both TBL and The End. The clues in the poster and book are simply giveaways about how Book the Thirteenth ends and hints as to the existence of multiple Beatrices, not anything more subtle than that.

Sept 8, 2019 13:03:37 GMT -5 Foxy said:

I think TE said the kids finally accepted that their parents had died, which is quite a fitting way to end a series, because acceptance is the final stage of grief.

'They cried for the world, and most of all, of course, the Baudelaire orphans cried for their parents, who they knew, finally, they would never see again.' (p. 307)