Using get_template_part to retrieve a template file based on current post type

There is built in support for that, kind of. While I don’t see anything wrong with your code, try naming your files on the form archive-mytype.php, content-mytype.php etc. and then call get_template_part like this:

<!--  This will result in including parts/content-mytype.php -->
<?php get_template_part('parts/content', get_post_type( $post )); ?>
<p id="t3-splash-title"><?php $post_type = get_post_type_object( get_post_type($post) ); echo $post_type->labels->singular_name ; ?></p>