You set "this" to a variable in the parent function and then use it in the inner function.

var simpleClass = function () {         
    this.status = "pending";     
    this.target = jqueryObject;     

    var parent = this;

    this.updateStatus = function() {         
            this.jqueryObject.fadeOut("fast",function () {            
                parent.status = "complete"; //this needs to update the parent class          
            });     
        }; 
    }; 

I will post this answer to this old question anyway as no one yet posted this before.

You can use the bind method on your function calls to define the scope which this belongs to.

https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Function/bind

Normaly everytime you create a method - this belongs to the current scope of the function. Variables from scope2 can't see variables from scope1.

e.g.

function(){
    // scope 1
    this.baz = 'foo';

    function(){
        // scope 2
        this.baz // not defined
    };
};

with the bind method you can define the scope from this inside the function. So using .bind(this) you're telling the called function that their own scope from this is referred to the scope of the parent function, like:

function(){
    // scope 1
    this.baz = 'foo';

    function(){
        // scope 1
        this.baz // foo
    }.bind(this);
};

so in your case, this would be an example using the bind method

var simpleClass = function () {    
    this.status = "pending";
    this.target = jqueryObject;
    this.updateStatus = function() {
        this.target.fadeOut("fast",function () {
           this.status = "complete"; //this needs to update the parent class 
        }.bind(this));
    }.bind(this);
};

Use an Arrow Function

An arrow function does not have it's own this. The this value of the enclosing lexical scope is used; arrow functions follow the normal variable lookup rules. So while searching for this which is not present in current scope they end up finding this from its enclosing scope.

Normal function syntax

function(param1, param2) {}

Arrow function syntax

(param1, param2) => {}

Usage

const simpleClass = function () {    
    this.status = "pending";
    this.target = jqueryObject;
    this.updateStatus = function() { 
        this.target.fadeOut("fast", () => { // notice the syntax here
           this.status = "complete"; // no change required here
        });
    };
};

Using an Arrow function within a ECMAScript 2015 Class

class simpleClass {

    constructor() {
        this.status = 'pending';
        this.target = jqueryObject;
    }

    updateStatus() {
        this.target.faceOut('fast', () => {
            this.status = "complete";
        });
    }
}

const s = new simpleClass();
s.updateStatus();

Described code works only in modern browsers.