With minimal changes to your existing code, a (untested) threaded solution might look like:

#!/usr/bin/perl -w
use strict;
use threads;
use threads::shared;

use LWP::UserAgent;
use HTTP::Request;

print "Working...\n";

my $ua = LWP::UserAgent->new;
$ua->agent("Mozilla/4.0 (compatible; MSIE 5.0; Windows 98; DigExt)");
$ua->timeout(15);

open URL_PLANETS, '<', "url_planets.txt"  or die $!;
my @urls = <URL_PLANETS>;
close(URL_PLANETS);
chomp @urls;

open NAMES, '>>', 'planet_names.txt' or die $!;

my $mutexStdout :shared;
my $mutexFile   :shared;
my $running     :shared = 0;

foreach my $planet (@urls)  {
    async {
        { lock $running; ++$running; }
        { lock $mutexStdout; print "Downloading: " , $planet , "\n" };
        my $req = HTTP::Request->new(GET => $planet);
        my $response = $ua->request($req);
        my $content = $response->content();
        
        lock $mutexFile;
        print NAMES $content =~ m[Rotations<i>(.*)</i>]m,"\n";
        { lock $running; --$running; }
    }->detach;
    sleep 1 while $running > 10;
}

sleep 1 while $running; ## Let the last 10 finish.
close(NAMES);
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Updated: misc++