You need to use SUPER. See perlobj. However, You're doing it wrong by putting a reference to parent in the child object. The whole point of inheritance is to make the parent's methods available to the child. See the second example.

Example 1: Your current code fixed up (as a monolithic file):

ackage MyParent;

use strict;
use Carp qw(carp croak);

our $VERSION = .01;

sub new {
    my $class = shift;
    my $self = bless {
      something => "foo",
    }, $class;
    return $self;
}

sub parent_nifty_method1 {
  my $self = shift;
  print $self->{something};
}

sub parent_nifty_method2 {
  my $self = shift;
  #... pah ...
}

package MyParent::MyChild;

use strict;
use base qw(MyParent);

our $VERSION = .01;

sub new {
    my $class = shift;

    my $parent = $class->SUPER::new;

    my $self = bless {
      PARENT => $parent,
    }, $class;

    return $self;
}

sub child_method_a {
   my $self = shift;
   #read parent variable, not change it
   print $self->{PARENT}->{something};
}

sub child_method_b {
   my $self = shift;
   #... incredibly cool stuff ...
}

package main;

#use MyParent;
#use MyParent::MyChild;
#use strict;

my $parentobject = MyParent->new();
$parentobject->parent_nifty_method1;
#prints: foo

#Now I want to access something or other in the MyChild, doing it this
+ way (which is what inheritance means to me)
my $childobject = MyParent::MyChild->new;

$childobject->child_method_a;
#Prints: foo

1;
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Example 2 - redefined MyParent::MyChild that actually inherits MyParent and builds on it:

package MyParent::MyChild;

use strict;
use base qw(MyParent);

our $VERSION = .01;

sub new {
    my $class = shift;
    my $parent = $class->SUPER::new;

    # extend blessed reference $parent with a new field
    $parent->{another_something} = "bar";
    my $self = bless $parent, $class;

    return $self;
}

sub child_method_a {
   my $self = shift;
   #read parent variable, not change it
   $self->parent_nifty_method1;
}

sub child_method_b {
   my $self = shift;
   print $self->{another_something};
}

package main;

#use MyParent;
#use MyParent::MyChild;
#use strict;

my $parentobject = MyParent->new();
$parentobject->parent_nifty_method1;
#prints: foo

#Now I want to access something or other in the MyChild, doing it this
+ way (which is what inheritance means to me)
my $childobject = MyParent::MyChild->new;

$childobject->child_method_a;
#Prints: foo

$childobject->parent_nifty_method1;
#Prints: foo

$childobject->child_method_b;
#Prints: bar
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