Override privileged method of base class

function BaseClass() {
    var map = {};
    this.parseXML = function(key, value) {
        alert("BaseClass::parseXML()");
        map[key] = value;
    }
}

function ChildClass() {
    BaseClass.call(this);
    var parseXML = this.parseXML;
    this.parseXML = function(key, value, otherData) {
        alert("ChildClass()::parseXML()");
        parseXML.call(this, key, value);
    }
}

ChildClass.prototype = new BaseClass;

var a = new ChildClass();
a.parseXML();

Live Example

Basically you cache the privileged method (which is only defined on the object) and then call it inside the new function you assign to the privileged method name.

However a more elegant solution would be:

function BaseClass() {
    this._map = {};
};

BaseClass.prototype.parseXML = function(key, value) {
    alert("BaseClass::parseXML()");
    this._map[key] = value;
}

function ChildClass() {
    BaseClass.call(this);
}

ChildClass.prototype = Object.create(BaseClass.prototype);
ChildClass.prototype.parseXML = function(key, value, otherData) {
    alert("ChildClass()::parseXML()");
    BaseClass.prototype.parseXML.call(this, key, value);
}

var a = new ChildClass();
a.parseXML();

Live Example

Also bonus implementation using pd