Custom Attribute:

input | data-parsley-date_is_after_or_equal_target_date | Start Date Field

JavaScript Field:

// Add new validation
window.Parsley.addValidator( 'date_is_after_or_equal_target_date', {
  requirementType: 'string',
  validateString: function( current_date_field_value, target_date_field_id ) {
		// Check if field exists
    if ( formyoula.form_fields[ target_date_field_id ] ) {
      // Get target date field value
      var target_date_field_value = new Date( formyoula.form_fields[ target_date_field_id ].get( 'value' ) );
      // Compare date fields
      var is_after = moment( current_date_field_value ).isAfter( target_date_field_value, 'day' );
      var is_equal = moment( current_date_field_value ).isSame( target_date_field_value, 'day' );
      // Return validation
      return is_after || is_equal;
    } 
  },
  messages: {
    en: 'End Date should be after or equal to Start Date'
  }
} );

Example:

Form Template for import:

[Example Form] Date Is After Or Equal Target Date.json