javascript accepting multiple date format

I should preface this post by saying that I am a very elementary developer with a generic IS degree. Without going into too much detail, I was given a moderately large web application from an interning software engineer to support an enhance if need be. It was written primarily in Python, JavaScript and HTML5 and utilizes a Google Map API to visually represent the location and uses of given inputs. This leads me to my question.

There is a date picker modal that the application/user utilizes. They pick a START and END date, in the default format YYYY-MM-DD (if the user does not use that exact format (i.e. 2015-09-29) the date picker will not work), and the application then goes to the DB and picks the given inputs between those dates and represents them on the map. I have been told that, for usability, I have to make the program accept multiple date formats (i.e. September 29 2015, 09-29-2015, 9-29-2015, 9/29/2015). How would I go about doing this?