Custom parent for a taxonomy

Generally, taxonomies inherit the root page of the post type they are most ‘strongly’ associated with (this is a nebulous simplification of what’s happening). In 5.3.0, the behavior behind this changes slightly so that if the taxonomy archive is restricted to a single post type, it will use that post type.

However, this doesn’t get you what you want in this case. Ultimately, it looks like you need a root page for a taxonomy. This is not something that Breadcrumb NavXT currently supports, though it is possible to add via an extension plugin (hook into bcn_after_fill, inject appropriate breadcrumb(s) into the passed in trail).

This is the code I used to fix the issue – any recommendations on what should be done different?

function fixBreadcrumbsForPrograms($trail) {
    global $wp_query;
    // This is a hacky way to fix the breadcrumb for Program pages
    $programsPage = 87; // Hard-coded pageid

    // We only want to edit the breadcrumb on program taxonomy pages
    if($wp_query->query_vars['taxonomy'] == 'program') {
        $title = get_the_title($programsPage);
        $url = get_the_permalink($programsPage);

        $trail->trail[1]->set_title($title);
        $trail->trail[1]->set_url($url);
    }
}
add_action('bcn_after_fill', __NAMESPACE__.'\\fixBreadcrumbsForPrograms');

If it seems to work, then it is a way of doing what you want. The only thing I’d normally recommend is instantiating a new instance of bcn_breadcrumb with the properties you desired (title, url, etc) and injecting it where you wanted in the array rather than resetting a few properties.