I would like to know if a permit is required to hold a rummage sale in a public parking lot?

It is difficult to provide a definitive answer to this question since vital information is missing from this question.  

First, if the event is to be held in an incorporated city, contact that city for an answer to your question.  If the event is to be held in an unincorporated area of the County, contact the County planning department to pose your question.  Be prepared to give an exact address for the site.  Contact the planning department by phone at:  913-715-2200.