How to add a new child category via an SQL statement?

Insert taxonomy terms via a raw SQL query is hard and discouraged, use the core wp_insert_term function, instead.

Assuming you have an array of continent/countries terms:

$countries_list = array(
  'Europe' => array(
    'Austria', 'France', 'Italy', 'Germany', 'Spain', 'United Kingdom'
  )
);

foreach ( $countries_list as $continent => $countries ) {
  // get term object for continent
  $parent = get_term_by( 'slug', sanitize_title( $continent ), 'category'  );
  if ( empty( $parent ) ) {
    // continent term doesn't exist, let's create it
    $parent_a = wp_insert_term( $continent, 'category' );
    if ( ! is_array( $parent_a ) ) continue;
    // get term object for the just created continent term
    $parent = get_term( $parent_a->term_id );
  }
  // loop countries to add the related terms
  foreach ( $countries as $country ) {
    // do nothing for current country if the related term already exists
    // see http://codex.wordpress.org/Function_Reference/term_exists
    if ( term_exists( sanitize_title( $country ), 'category', $parent->term_id ) ) {
      continue;
    }
    // insert the country term using continent as parent
    wp_insert_term( $country, 'category', array( 'parent' => $parent->term_id ) );
  }
}

Of course, this is a very resource-expensive code, and even if code does checks to avoid terms are added more than once, you should be sure this code run only once, to avoid serious performance issues.

To setup a run-once script, have a look at this Q/A.