I'm sure the cops will play along for a few days or a week or so...makes them look evil if they immediately come out publicly with doubts about the story, the parents go ballistic and go on "Good Morning America", etc.
No matter what, the girl actually is missing, so they can just look for her. And I'm sure they're suspicious about the story and working quietly behind the scenes on their suspicions, while the public search for the girl goes on.