In DOM parser is used at server side in php using domxml.
While in AUX parser is used at client side in javascript

No. XMLHttpRequest is only part of the Ajax equation. XMLHttpRequest is the technical component that makes the asynchronous server communication possible; Ajax is our name for the overall approach described in the article, which relies not only on XMLHttpRequest, but on CSS, DOM, and other technologies.

DOM it is a hierarchical model that allows developers to manipulate XML documents easily.ie maintained by the W3C DOM Workgroup that defines an application independent mechanism to access, parse, or update XML data.