Is it possible to get render a field for a paragraphs field in the parent twig template
in your node.html.twig file you can traverse up the element array like this 

{{node.field_parent.entity.field_paragraph.entity.field_text.value}}


where 


field_parent.entity is the parent paragraph
field_paragraph.entity is the child paragraph attached to parent
field_text is a text field attached to child paragraph


OR inside your paragraph.html.twig 

{{paragraph.field_paragraph.entity.field_text.value}}


NOTE: its actually better to put this code in a paragraph--parent.html.twig file otherwise this template is going to be used also for the child text paragraph which does not have field_paragraph and its going to lead to error.