Unless the OP edited the code, his example does not have a closure. He alluded to tracking the exit codes via a HASH but that sounded more like a global than a closure. I believe the OP may be looking for someway to update his database inside the run_on_finish command -- in that case he could just close over the database handle instead of some data structure which could be manipulated *after* the run of all the children.

Update: ... forgot an (untested) example:

my $dbh = create_dbh_handle();
my $code = create_closure( $dbh ); 
.... 
my $pm = Parallel::ForkManager->new( $MAX_NR_PROCESSES );
   $pm->run_on_finish( $code ); 

foreach ... {
  my $pid = $pm->start and next;
  ...
  $pm->finish; # Terminates the child process
}

sub create_closure { 
  my( $dbh ) = @_; 
  return sub { 
    my ($pid, $exit_code, $ident) = @_; 
    my $sql = "insert into table values( $pid, $exit_code )";
    $dbh->do( $sql );
  }
}
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Unless the OP edited the code, his example does not have a closure.

He didn't edit anything.

sub { my ($pid, $exit_code, $ident) = @_; ... } is a closure that captures $MAX_PROCESSES, $pm, and anything else he declares there (like $stats or $dbh).

You didn't create a closure, you just refactored the code. As such, your solution still doesn't work. See Re^3: Parallel::ForkManager run_on_finish exit code handler for a description of the actual problem and a solution for it.