SELECT 
  category_id, 
  parent_id 
FROM 
  cscart_categories 
WHERE 
  parent_id IN(
    18559, 18560, 18561, 18562, 18563, 18564, 
    18565, 18566, 18567, 18568, 18569, 
    18573, 18580
  ) 
  AND category_id != 1

Query time 0.00005

JSON explain

{
  "query_block": {
    "select_id": 1,
    "nested_loop": [
      {
        "table": {
          "table_name": "cscart_categories",
          "access_type": "range",
          "possible_keys": ["PRIMARY", "parent", "p_category_id"],
          "key": "parent",
          "key_length": "3",
          "used_key_parts": ["parent_id"],
          "rows": 22,
          "filtered": 100,
          "index_condition": "cscart_categories.parent_id in (18559,18560,18561,18562,18563,18564,18565,18566,18567,18568,18569,18573,18580)",
          "attached_condition": "cscart_categories.category_id <> 1"
        }
      }
    ]
  }
}