You don't need to call the Parser class. If you load the package it works automatically.
The theme class works with Views, if you want to call a template engine with the parser class you'll have to tell the parser which engine to use. You do that using the extension, which in case of Smarty is ".smarty" (because ".tpl" was already used. If you want to use that, copy the parser config to app/config, and modify it). By default it uses ".php", using "default" is the same as using "default.php".
So it will become \Theme::instance()->set_template('default.smarty'); The same for all partials.