WWW::Mechanize defines get() like this:

So you could simply do this:

my $mech = ....;

for my $page_number (1 .. 1_000_000) {
    
    my $uri = "siteone_dot_com?show_subsite=$page_number";
    my $response = $mech->get($uri);


}
[download]

But it also appears you might be able to do something like this:

my $mech = ....;

my $url = "siteone_dot_com";
my $param = 'show_subsite';

for my $value (1 .. 1_000_000) {
    
    my $response = $mech->get($url, $param => $value);


}
[download]

Experiment and see if they both work.