"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. ... 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 fulfil. I am your daughter, sir."
-- Delphi to Harry/Voldemort (CC4.11)
A prophecy from an unknown source stating: “When spares are spared, when time is turned, when unseen children murder their fathers: then will the Dark Lord return” (CC3.19, CC3.21).
- The prophecy was given to Delphi, child of Voldemort and Bellatrix Lestrange, by Rodolphus Lestrange after he was released from Azkaban (CC4.11).
- Cedric Diggory was thought to be the "spare" mentioned in the prophecy, since Voldemort ordered Peter/Wormtail to "kill the spare" in the Little Hangleton graveyard, meaning that only Harry was needed to help the Dark Lord be reborn, and Cedric was a throw-away who unfortunately grabbed the portkey from the Maze at the same time (GF32).
- Delphi had hidden the prophecy inside a hissing oil lamp at St. Oswald's Home for Old Witches and Wizards, where it was found by Ginny Weasley. The hissing noise turned out to be Parseltongue language, and when released the walls became covered with snakes and the words of the prophecy were written over them in fluorescent green (CC3.21). Delphi's evil intentions also appeared written on the ceiling: "I will rebirth the Dark. I will bring my father back" (CC3.21).
- The reason Delphi confunded Amos Diggory into believing she was his niece, and recruited Albus and Scorpius to steal the Experimental Time-Turner was to "spare the spare" by going back in time to save Cedric and bring her father back to life (CC1.6).
- Albus Potter succeeded in "murdering his father" during one trip back with the Time-Turner. Cedric was humiliated at the Triwizard Tournament when Scorpius and Albus turned him into a balloon and floated him out of the water during the Second Task, thus sparing his life (CC2.20). Unfortunately, Cedric was so humilated that he later became a Death Eater, and killed Neville Longbottom at the Battle of Hogwarts, leaving one Horcrux available to keep Voldemort alive while Harry was killed instead. Harry's death in the past meant that his future son Albus disappeared, leaving only Scorpius to return to a world in which Voldemort ruled with his daughter Delphi at his side, Draco was Head Auror, and Snape, Hermione, and Ron tried to fight the system in secret from under the Whomping Willow (CC3.6, CC3.7).
One could say that Delphi fulfilled her own prophecy when she came to Godric's Hollow as Voldemort's "unseen" child who stood by as he was "murdered" by Baby Harry on Halloween night. She wanted to stop what was happening so her father would survive and become stronger since the Cedric plan did not work out, but Harry disguised himself as Voldemort and kept her from interfering with his own prophecy about the "one with the power to vanquish the Dark Lord" - himself as a baby. It all gets very complicated - did all of this happen simultaneously in the regular canon, just outside the home of James and Lily Potter? ~ SIP
Although the first time we hear the prophecy is when Delphi forcibly takes Albus Potter and Scorpius Malfoy back in time to the Third Task of the Triwizard Tournament (CC3.19), the "big reveal" is when it is discovered in her room at St Oswald's Home, the words pulsating evilly around the walls (CC3.21).
From the Web
Pottermore: Harry Potter and the Cursed Child
The theatre production opened in the West End of London at the Palace Theatre on 31 July 2016: Harry Potter and the Cursed Child, Parts One and Two