There are several options for parsing HTML, for something like this it seems easiest to use Mojo::DOM:

my $dom = Mojo::DOM->new($html);
my $values = $dom->find('select#jumpMenu2 > option')
    ->map(attr=>'value')->compact;  # returns a Mojo::Collection

for my $v (@$values) {
    print "<<$v>>\n";
}
[download]

Update: But it's also possible with WWW::Mechanize directly, see my other reply.