Consider passing what you call "the parent" around to each object that implements an action? So far you've been somewhat unclear about what you actually need, so it is somewhat hard to come up with good names for the parts that make sense in your context.

Maybe using "configuration" as name for what you call "parent" and "action" as name for what you call "child" would help?

my $cfg = My::Config->new_from_file('nysus.yml');
$cfg->process( $myfile );

package My::Config;
use Module::Pluggable require => '1',
                      sub_name => 'actions',
                      instantiate => 'new',
                      ;

sub My::Config::process( $self, $filename ) {
    for my $action ($self->actions) {
        $action->operate( $self, $filename );
    };
};
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I don't think that passing data around in $self should be an overarching design principle.

Most of the code above is from Module::Pluggable, which still feels very applicable from what you've described so far and from the problems you encounter.

Maybe if you can elaborate more on the concrete problem you're trying to solve and the problems you encounter, we can give you more concrete advice?