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SELECT 
  category_id, 
  parent_id 
FROM 
  jmj_categories 
WHERE 
  parent_id IN(
    109, 116, 88, 79, 80, 117, 110, 111, 81, 
    118, 119, 112, 120, 113, 114, 121, 115, 
    122
  ) 
  AND category_id != 1

Query time 0.00060

JSON explain

{
  "query_block": {
    "select_id": 1,
    "cost_info": {
      "query_cost": "29.42"
    },
    "table": {
      "table_name": "jmj_categories",
      "access_type": "ALL",
      "possible_keys": [
        "PRIMARY",
        "parent",
        "p_category_id"
      ],
      "rows_examined_per_scan": 120,
      "rows_produced_per_join": 30,
      "filtered": "25.00",
      "cost_info": {
        "read_cost": "23.42",
        "eval_cost": "6.00",
        "prefix_cost": "29.42",
        "data_read_per_join": "116K"
      },
      "used_columns": [
        "category_id",
        "parent_id"
      ],
      "attached_condition": "((`jmj_test`.`jmj_categories`.`parent_id` in (109,116,88,79,80,117,110,111,81,118,119,112,120,113,114,121,115,122)) and (`jmj_test`.`jmj_categories`.`category_id` <> 1))"
    }
  }
}

Result

category_id	parent_id
88	79
109	80
116	81
110	80
117	81
111	80
118	81
112	80
119	81
113	80
120	81
114	80
121	81
115	80
122	81

category_id	parent_id
88	79
109	80
116	81
110	80
117	81
111	80
118	81
112	80
119	81
113	80
120	81
114	80
121	81
115	80
122	81

category_id	parent_id
88	79
109	80
116	81
110	80
117	81
111	80
118	81
112	80
119	81
113	80
120	81
114	80
121	81
115	80
122	81

category_id	parent_id
88	79
109	80
116	81
110	80
117	81
111	80
118	81
112	80
119	81
113	80
120	81
114	80
121	81
115	80
122	81