No. You do not have access to environment variables on the about:config page.

You can get that value via the Tools > Error Console if necessary. Copy and Paste this code in the Code field and click evaluate.

Components.classes["@mozilla.org/process/environment;1"].getService(Components.interfaces.nsIEnvironment).get('USERNAME');

With that code snippet, can the response be injected into the browser.cache.disk.parent_directory setting?

An extension should be able to do that quite easily.
Try this code in the Tools Error Console.
You can create an extension to make the change.

const Cc = Components.classes, Ci = Components.interfaces;
var PB = Cc["@mozilla.org/preferences-service;1"].getService(Ci.nsIPrefService).getBranch("");
var PN = "browser.cache.disk.parent_directory";
var curVal = PB.getCharPref(PN);
var userName = Cc["@mozilla.org/process/environment;1"].getService(Ci.nsIEnvironment).get('USERNAME');
newVal = curVal.replace(/%USERNAME%/i, userName);
PB.setCharPref(PN, newVal);

See also:

• https://developer.mozilla.org/en/nsIPrefBranch
• https://developer.mozilla.org/en/NsIEnvironment

By "create an extension", do you mean just paste that code into a XPI?