That's correct. First (FWIW) I can confirm your observation (SuSE 11.1, 2.6.27.7-9-pae, perl 5.10.0).

The problem seems to be that your daemon exec's from within the signal-handler (presumably inheriting a blocked signal mask). After applying a quick and dirty work-around, the program reloaded as expected multiple times. The work-around was to set a global variable $PLEASE_RELOAD=1; (initialised to 0) within the HUP-handler and to incorporate something like relaunch( $runDir, $pidFile, $exeFile ) if $PLEASE_RELOAD; into hupMain() allowing to exec outside the HUP signal handler.

This is ugly, but it seems to locate the problem somewhere near the HUP handler...
Another hint: you used a variable called $parentPid to save the children/daemons PID.

Update: Maybe this snipped from perlipc provides a useful hint?

         # POSIX unmasks the sigprocmask properly
         my $sigset = POSIX::SigSet->new();
         my $action = POSIX::SigAction->new('sigHUP_handler',
                                            $sigset,
                                            &POSIX::SA_NODEFER);
         POSIX::sigaction(&POSIX::SIGHUP, $action);

         sub sigHUP_handler {
             print "got SIGHUP\n";
             exec($SELF, @ARGV) or die "Couldn't restart: $!\n";
         }
[download]

Update: Hm, forget to note: SIGHUPSIGTERM (Argl. Sorry, confused that, AM is right, see below.) is send by the OS on shutdown (usually followed by SIGKILL a little bit later) - this would relaunch your daemon during shutdown. Maybe it is a better idea to use SIGUSR1 to reload the daemon?