To have a variable in a field template you need a field preprocess hook:

 use Drupal\paragraphs\ParagraphInterface;

 function mysubtheme_preprocess_field(&$variables) {
 
 $paragraph = $variables['element']['#object'];
 if ($paragraph instanceof ParagraphInterface) {
 if ($paragraph->getType() === 'sidebar_button_repeater') {
 if (!$paragraph->field_background_color->isEmpty()) {
 $variables['color'] = $paragraph->get('field_background_color')->value;
 }
 }
 }

 } 

If the field is in the parent paragraph of a nested paragraph:

 $parent = $paragraph->getParentEntity();

And get the field value from $parent:

 function mysubtheme_preprocess_field(&$variables) {
 
 $paragraph = $variables['element']['#object'];
 if ($paragraph instanceof ParagraphInterface) {
 if ($paragraph->getType() === 'child_type') {
 $parent = $paragraph->getParentEntity();
 if ($parent instanceof ParagraphInterface) { 
 if ($parent->getType() === 'parent_type') {
 if (!$parent->field_background_color->isEmpty()) {
 $variables['color'] = $parent->get('field_background_color')->value;
 }
 }
 }
 }
 }

 }