I think Beatrice Baudelaire is parentless because...
Kit Snicket, the original mother, died of the Medusoid Mycelium after giving birth.
Dewey, which in the TV show, him and Kit were together, but again, Dewey died due to the harpoon accident. In the books, Kit just said that he was a good man.
Count Olaf. Count Olaf and Kit Snicket were together for some time. But also, he died due to a harpoon shooting by Ishmael.
The Baudelaires. Kit told the Baudelaires to take care of Beatrice. But the thing is that they were on the Island, with no laws or government. We don't know if the Baudelaires got to the city, but we know Beatrice did. Violet wasn't 18 yet, so she probably coudn't adopt Beatrice.