The case against five teens accused of raping a young woman in a Brooklyn playground is crumbling as investigators cast doubt on her father’s story, law-enforcement sources told The Post Wednesday.

The 18-year-old victim’s dad initially claimed he ran for his life when the teenagers held them up at gunpoint in the Osborn Playground in Brownsville at around 9 p.m. last Thursday, sources said.

But no gun has been found, and cellphone footage from the moments before the alleged rape shows a young woman sitting on the ground and grinning, while some of the teens laugh in the background.

Two of the suspects have told police the father was having sex with his daughter when they arrived — allegations police are checking out.

“The father’s initial story just doesn’t hold up to scrutiny,” a source said.

Four of the teens have been arraigned and remain behind bars. Travis Beckford, 17, the last of the suspects arrested, was arraigned Wednesday night. He was held in lieu of $2,500 bail.