JSON.parse() is part of the EcmaScript 5 standard - XML.parse() isn’t a thing. Kind of makes sense - parsing JSON is a pretty one-to-one with javascript objects, while XML doesn’t map as nicely (with attributes etc).
For parsing Java, I’m sure you can use whatever Enonic uses by invoking Java. After some digging, I found Enonic’s XML-parsing classes here:
An alternative can be to include your XML parser of choice in the Gradle build script and include it the same way.
A third alternative, in pure JS, is to use a webjar with a JS lib for parsing XML. Find whatever you need on webjars.org, and include it in the Gradle build script of your site. XML2JS looked good, but I’ve never used an XML parsing lib for JS, so I can’t really speak to it. This question on SO has some good suggestions as well.
You are trying to invoke something that doesnt exist in the api, com.enonic.xp.xml.parser is just a package, and the classes in there will probably not help you.
I’ve created a small test-project that should give you a pointer on how to setup a java based xml-parser: