Delphini received the prophecy from Rodolphus Lestrange.
DELPHI: I am from the future. The child of Bellatrix Lestrange and you. I was born in Malfoy Manor before the Battle of Hogwarts. A
battle you are going to lose. I have come to save you.
[HARRY/VOLDEMORT turns. She meets his eyes.]
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.
Harry Potter and the Cursed Child - Act 4, Scene 11
It's not really clear where he got the prophecy from, but there have been mentions on Pottermore that some parents visit "naming seers" who provide general predictions about a child's future.
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
This practice is becoming increasingly rare. Many parents prefer to
'let him/her find his/her own way', and dislike (with good reason)
receiving premature hints of aptitude, limitations or, at worst,
catastrophe. Mothers and fathers have often fretted themselves silly
on the way home from the Naming Seer, wishing that they had not heard
the Seer's predictions about their child's personality or future.
Naming Seers By J.K. Rowling