I would like to know how can I parse XML files and perform read write operations using Drupal 6.
There is no API for this in Drupal -- use the SimpleXML extension from PHP. It works wonders.
It depends on the specifics of what you need to do. Modules such as Import / Export API, Views Datasource, XML Parser, and others may be useful, depending on what exactly you need to accomplish.
Also, this XML Parser for the Feeds module looks promising.
Drupal is a PHP application. There is no specific XML API in Drupal and any PHP one can be used in a custom module for a specific user case or feature. There is a large collection of XML APIs in PHP. If you have experience with jQuery, look also at QueryPath and the QueryPath module for Drupal.
You will need to create a module to embed your custom PHP code in your Drupal site.
As pointed by Matt V., there is a lot of modules in Drupal that deal with XML. Some may fit your needs and avoid you to write any custom code.