Change URL structure of subcategory archive pages

I have my permalink structure for posts set up like this: /%category%/%postname%/

This generates URLs like this: mysite.io/category/sub-category/post-name

Thats all fine.
However, the archive page URLs for subcategories are like this: mysite.io/sub-category

Instead, the URLs for subcategories should be like this: mysite.io/category/sub-category

This category type is actually a custom taxonomy registered by the parent theme. I don’t think it is possible to use the permalink settings in the admin dashboard to change the URL structure of this but perhaps there is something I could add to the functions.php to alter the registered taxonomy?

I have found this page in WordPress codex and think it might be useful for finding a solution but my WordPress knowledge seems way too limited to understand how:

‘hierarchical’ – true or false allow hierarchical urls (implemented
in Version 3.1) – defaults to false

Function Reference/register taxonomy

Thanks in advance

You answered your own question yourself.

You want links to custom categories to look like this

{taxonomy_slug}/{parent_term}/{child_term}/{grandchild_term}/

so you should pay attention to two parameters in the register_taxonomy() arguments: hierarchical and rewrite.

$args = [
    'hierarchical' => true,      // <-- term may have a parent
    'labels'       => $labels,
    'rewrite'      => [
        // hierarchical urls, defaults to FALSE
        'hierarchical' => true,  // <-- 
    ]    
];

Your custom taxonomy is created by the parent theme, so to change it, use the register_taxonomy_args filter:

add_filter( 'register_taxonomy_args', 'se344007_mytax_args', 10, 2 );
function se344007_mytax_args( $args, $taxonomy )
{
    if ( 'mytax' !== $taxonomy ) {
        return $args;
    }    
    // it looks like it's already set up by parent theme
    // $args['hierarchical'] = true;

    if ( !is_array($args['rewrite']) )
        $args['rewrite'] = [];
    $args['rewrite']['hierarchical'] = true;

    return $args;      
}

When you register custom taxonomy, the default link to term (custom category)
is {taxonomy_slug}/{child_term_slug} even if taxonomy is hierarchical, because
by default, the created links are not hierarchical ( $args[‘rewrite’][‘hierarchical’] = false ).