WordPress using get_term to retreive slug not working as expected

I might be completely missing what you want here, but it sounds like you’re saying you want the parent category’s slug to echo out, rather than the parent category’s human readable name.

Rather than this:

echo $parent->name;

You just need this instead:

echo $parent->slug;

You can see a list of the properties here that are returned by the get_term() function. You can also always use PHP’s print_r() function on the term object (so, print_r( $parent ); in this case) to see all of the properties you have available to you.