Semantic quibble: these aren't threads, they're processes. There's a big difference with regard to synchronicity and shared resources.

That said, what happens if you close the connection in the parent after you've forked off a child? If one process still has a connection open to the client, how does the client know it's finished reading? You might be able to get away with this:

while (my $client = $server->accept())
{
    # as normal
}
[download]

This is untested, though, and I haven't looked in a networking book to be sure.