An update with references. The BBC history magazine has a review of a book "The Cerne Giant: an antiquity on trial" Oxbow Books 1999, from Bournemouth University. The perpetrator is claimed to be MP Denzil Holles, originally one of the 5 whom Charles I tried to arrest, but later an enemy of Cromwell. A 1751 "History of Dorset" by the Rev. John Hutchens attributes the giant to Holles, who owned the hillside through his second wife's dowry. Katherine Barker of Bournemouth sees the image as an allegorical 17th century political cartoon, and appropriate to that period. Cromwell had been portrayed as Hercules by his admirers, the club could represent oppressive rule, and the other obvious feature a reference to Puritanism.

I'm not so sure about the latter - did the 17th century see puritanism as so much to do with sexual repression as we do? And is it an appropriate jibe against Cromwell?

Also, the joke would not have worked if it were kept private, and once public, surely Cromwell's supporters would have razed it, or obliterated it somehow.

Penny