To get your idea working, you need to look at these lines of code:

# (1) your working example
$res = $ua->request(HTTP::Request->new(GET => $url),
                      sub {$p->parse($_[0])});
# compared with (2)
$res = $ua->request($req)->as_string,
            sub {$p->parse($_[0])};
[download]

Looking carefully at the bracketing in the second line, it appears it should be more like:

Your version was not passing the callback to call LinkExtor to the UserAgent request method. The call signature for the request method in the form you want it is:

where $request is a HTTP::Request object and &callback is a sub or whatever.