Classical question, there should be a Wiki article on this topic...
The usual way is to pass an instance of the PApplet ( this in the main code) to your class, usually in the constructor (as done for XMLElement, precisely). You can then use this instance to pass to other classes.
Now, I was curious to see why XMLElement needed a PApplet instance. It appears that, currently at least, it is used only in save() (probably to get the sketch's location). So it you don't use this function, you can just pass null.

Thanks for your response, but when I pass it null, I get a null pointer exception, even if I don't use save.  I suppose I could pass down through my classes references to the PApplet, but I would rather not if I can avoid it.

Ah, indeed, it is also used to create a reader: init(sketch.createReader(filename));
Another constructor takes directly only the reader, so you can call it instead: new XMLElement(pa.createReader(filename)); for example, but you still need a reference to the PApplet (pa) or another way to create a reader.