It looks to me like your threaded version has a single process that's trying to open the same file 4 times and read from 4 different file positions?

In the routine thread_process(), the list of files is pushed onto a queue. Each of the threads then pull (different) filenames from that queue:

sub thread_process {
    my $args = shift;
    my @threads;
    my $queue = Thread::Queue->new();
    for my $p ( 1 .. $PARALLEL ) { 
        push @threads, threads->create( sub {
            while( my $file = $queue->dequeue) { 
                process($file, "Thread $p of $PARALLEL");
        }})
    }
    $queue->enqueue( @$args, (undef) x $PARALLEL );
    $_->join for @threads;
    return 1;
}
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So no. They are not reading from the same file.