Police cordoned off the park and had forensic officers on the scene following the claim and put out an appeal to find the "suspect," described as aged between 40 and 45, 6ft tall, of large build and with wrinkles on his face, particularly around the cheeks.
But the following evening police said they were no longer appealing for information and revealed the woman who had made the allegation had been formally cautioned for wasting police time.
The bogus claim provoked fury on Facebook where people questioned why she had been cautioned rather than charged.
One post said: "The woman should have been named, shamed and charged. This is not only wasting police time but could have had a devastating effect on an innocent man who could have been subjected to the nightmare of being falsely accused.
"His name would have been all over the newspapers and this type of crime could have stigmatised him for ever."
Speaking from her Whitehall area home the woman's mother said her daughter had suffered since the false claim had come to light. She added that it was "a cry for help" related to an incident against her a couple of years ago.
A Kent Police spokesman said: "Due to the extremely sensitive nature of rape allegations we take all reports seriously and have a duty of care to fully investigate them.
"Kent Police always encourages victims of rape to come forward and we have specialist teams who can provide them help and support.
This false allegation generated many hours of police time that could have been spent investigating a genuine crime and the person who admitted making it was formally cautioned.
"Making a false allegation is a serious offence and can, in some circumstances, result in a prison sentence."