People who definitely knew before 2017
Delphini's mother, Bellatrix and her husband Rodolphus were aware
Delphi: It was Rodolphus Lestrange, Bellatrix’s loyal husband, who on return
from Azkaban told me who I was and revealed the prophecy he thought I
was destined to fulfill. I am your daughter, sir.
Euphemia Rowle (probably)
Delphi: She used to say it was crying because it could see I was going to come to a sticky end. She didn’t like me much. Euphemia
Rowle . . . she only took me in for the gold.
Whoever made the prophecy about her
A recent Pottermore article mentions "Naming Seers".
A certain sector of magical society, however, follows the ancient
wizarding practice of consulting a Naming Seer, who (usually for a
hefty payment of gold) will predict the child's future and suggest an
And that's pretty much it. There's no indication that Voldemort was aware that he had a daughter.
Note that this appears to be a retcon by those that wrote CC since there's no mention in the books (by the Death Eaters who were present at the Malfoy's mansion) of what should have been a visible late-stage pregnancy. This could mean, theoretically, that Delphini's father was simply lying to her about her parentage for reasons of his own.