Dr. Strangelove was released before Fail Safe. The story goes that they were both being filmed around the same time and Kubrick used his pull with the studio to make sure Fail Safe was released later in the year.

Seems a really odd thing to insist your parody is released before the movie it’s parodying. And I don’t think there were all that many movies about the terror of nuclear war until after the Cuban missile crisis. It takes a couple of years to make a movie and Dr. Strangelove came out less than two years after the Cuban Missile Crisis, so it was pretty much the first of it’s kind.

Seems to me like Dr. Strangelove is a black comedy, not a parody.