Is there any way to get all the name or slug of template parts used in a page?

Question

You could filter the include paths returned by get_included_files by removing any files from the list that are not in your theme (and/or child theme) directory:

function get_theme_includes() {

    $includedfiles = get_included_files();
    // normalize theme paths for matching
    $styledir = str_replace("\\","https://wordpress.stackexchange.com/",get_stylesheet_directory());
    $templatedir = str_replace("\\","https://wordpress.stackexchange.com/",get_template_directory());

    $i = 0; // loop included files
    foreach ($includedfiles as $includedfile) {
        // normalize include path for match
        $includedfile = str_replace("\\","https://wordpress.stackexchange.com/",$includedfile);
        // check if included file is in stylesheet directory
        if (substr($includedfile,0,strlen($styledir)) != $styledir) {
            // if stylesheet is same as template, not a child theme
            if ($styledir == $templatedir) {unset($includedfiles[$vi]);}
            else {
                // check if included file is in template directory
                if (substr($includedfile,0,strlen($templatedir)) != $templatedir) {unset($includedfiles[$i]);}
                else {
                    // strip template directory from include path
                    $pathinfo = pathinfo(str_replace(dirname($templatedir),'',$includedfile));
                    // add filename.php => pathinfo array to the template array
                    $themeincludes[$pathinfo['basename']] = $pathinfo;
                }
            }
        } else {
            // strip stylesheet dir from include path
            $pathinfo = pathinfo(str_replace(dirname($styledir),'',$includedfile));
            // add filename.php => pathinfo array to the template array
            $themeincludes[$pathinfo['basename']] = $pathinfo;
        }
        $i++;
    }
    return $themeincludes;
}

Depending on when this function is run however you would get different results, you can use this to your advantage to further remove early theme includes since those are likely not a page templates.

add_action('wp_loaded','check_theme_includes');
add_action('wp_footer','check_theme_templates');

function check_theme_includes() {
    global $themeincludes; $themeincludes = get_theme_includes();
}

function check_theme_templates() {
    global $themeincludes, $templateincludes;
    $templateincludes = get_theme_includes();

    // strip out already included theme files from template list
    foreach ($templateincludes as $template => $pathinfo) {
        if (array_key_exists($template,$themeincludes)) {
            if ($pathinfo['dirname'] == $themeincludes[$template]['dirname']) {
                unset($templateincludes[$template]);
            }
        }
    }

    // debug point
    // echo "<!-- INCLUDED TEMPLATES: "; print_r($templateincludes); echo "-->";

   // output template array for use by jquery/ajax
   echo "<script>var templatenames = new Array(); var templatepaths = new Array(); ";
   $i = 0;
   foreach ($templateincludes as $template => $pathinfo) {
       // optionally strip the .php extension
       $template = str_replace('.php','',$template);
       // output the template array key/value
       echo "templatenames[".$i."] = '".$pathinfo['filename']."'; ";
       echo "templatepaths[".$i."] = '".$pathinfo['dirname']."'; ";
       $i++;
   }
   echo "</script>";
}

Which for example would give something like this in your page footer:

Which would allow you to compare these arrays and add to them with jQuery/javascript callbacks.