JavaScript already has a Date.parse() method, why would we talk about date parsing?

Because in short the Date.parse() method implementation is dependent on the browser itself, so it could return a different result in chrome/firefox/ie like here**

Introducing Moment.js

Moment.js is a library for parsing, validating, manipulating, and formatting dates.

it has an excellent documentation, it has plenty of helpful examples

Parsing & Validating dates

var isValid = moment("12-252-1995").isValid() //returns false  
var date = moment("12-25-1995", "MM-DD-YYYY") //prase a date using a specify format, if the actual date is in a different format isValid() will return false  
var date = moment("12-25-1995", ["MM-DD-YYYY", "YYYY-MM-DD"]) //parse a date with a list of possible formats that it could matches

//using it as a wrapper of a date object
var day = new Date(2011, 9, 16);  
var dayWrapper = moment(day);  

Bonus for MVC users

ASP.NET returns dates in Json as /Date(1198908717056)/ or /Date(1198908717056-0700)/

This will happen often if you try to make an ajax call to an action and it returns a JsonResult that contains DateTime

 var date = moment("/Date(1198908717056-0700)/") // December 28 2007 10:11 PM