SELECT 
  category_id, 
  parent_id 
FROM 
  cscart_categories 
WHERE 
  parent_id IN(
    627, 556, 505, 565, 506, 512, 513, 557
  ) 
  AND category_id != 1

Query time 0.00015

JSON explain

{
  "query_block": {
    "select_id": 1,
    "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": 8,
      "filtered": 100,
      "index_condition": "cscart_categories.parent_id in (627,556,505,565,506,512,513,557)",
      "attached_condition": "cscart_categories.category_id <> 1"
    }
  }
}