The error code is in $!.
If a handle is set to non-blocking and there is no data present and a sysread returns undef, the error should be either EWOULDBLOCK or EAGAIN. Any thing else is a real error.
The error text you were seeing "Resource temporarily unavailable" was from EAGAIN.
sysread() ultimately calls the C function "read", see "man 2 read" for information about read errors includiung EAGAIN.

Setting $parent to non-blocking does not cause an error, did you mean $child ?

Another problem with your code is the sleep in the receiver section. If data arrives on the socket during that sleep, response will be delayed until that sleep completes. In general, there should be no sleeps in a receiver except for the timeout in can_read or select. This is why select() and IO::Select were invented!

In general, any attempt to use non-blocking should be avoided, and should require exceptional justification before it is allowed.

Following is a set of suggested "tweaks" for eliminating that problem and several others.

#!/usr/bin/perl

use diagnostics;
use strict 'subs';
use strict 'refs';
use Socket;
use IO::Handle;
use IO::Select;

my $child;      # filehandle to child process
my $parent;     # filehandle to parent process
my $pid;        # Process ID of child process

#       w r i t e L i n e
# Writes a buffer to the filehandle.
sub writeLine {
    my ($fh, $buf) = @_;
    print $fh $buf;
#   while( length $buf ) # alternate to yours, but single print is the
+ same
#     {
#     my $stat = syswrite $fh, $buf;
#     $stat and substr $buf, 0, $stat, '';
#     }
}                                               #writeLine()

socketpair($child, $parent, AF_UNIX, SOCK_STREAM, PF_UNSPEC)
  or die "socketpair: $!";
$child->autoflush(1);
$parent->autoflush(1);
if ($pid = fork()) {                    #parent
    close $parent  or die "close: $!\n";
    my $sel = IO::Select->new($child);
    my @handles;
    my $buf = '';
    while ($sel->count)
      {
#     print STDOUT time%100, ": polling child\n";
      for my $fh ( @handles = $sel->can_read(1) )
        {
        if( sysread $fh, $buf, 1e6, length $buf )
          {
          print STDOUT time%100, ": received <$1>\n" # because you may
+ get more
            while $buf =~ s/(.*)\n//;                # than one line a
+t a time
          }
        else
          {
          $sel->remove($fh);
          }
        }
      @handles or print STDOUT time%100, ": no input from child at thi
+s time\n";
      }
    my $stat = wait;
    die "wait returned $stat\n"  unless $stat == $pid;
    print STDOUT time%100, ": child reaped, parent exiting\n";
    exit 0;
    }
  else {
    die "cannot fork: $!"  unless defined $pid;
    close $child or die "close: $!\n";
    writeLine($parent, time%100 . ": child started\n");
    sleep 4;
    writeLine($parent, time%100 . ": child wrote again\nwith two lines
+\n");
    sleep 2;
    writeLine($parent, "E_O_F\n");
    close $parent or die "close: $!\n";          #causes termination
    print STDOUT time%100, ": child exiting\n";
    exit;
}
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EDIT: removed overlooked ->blocking call.

With the select, there's no point in making the handle non-blocking anymore.

That said, I don't understand why a non-blocking handle or select is being used at all.

The non-blocking call should have been removed, but was overlooked. It's gone now :)

sysread is a thin wrapper around the read system call. It would help you to read its documentation (the read(2) man page). From the "Errors" section,

EAGAIN or EWOULDBLOCK

The file descriptor fd refers to a socket and has been marked nonblocking (O_NONBLOCK), and the read would block. POSIX.1-2001 allows either error to be returned for this case, and does not require these constants to have the same value, so a portable application should check for both possibilities.

I would have used:

if (!defined($stat)) {
   if $!{EAGAIN} || $!{EWOULDBLOCK}) {
      ... No data at this time ...
   }

   die "sysread: $!\n";
}

if (!$stat) {
   ... EOF ...
)
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