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SELECT 
  category_id, 
  parent_id 
FROM 
  jmj_categories 
WHERE 
  parent_id IN(67, 68, 82) 
  AND category_id != 1

Query time 0.00875

JSON explain

{
  "query_block": {
    "select_id": 1,
    "cost_info": {
      "query_cost": "28.21"
    },
    "table": {
      "table_name": "jmj_categories",
      "access_type": "range",
      "possible_keys": [
        "PRIMARY",
        "parent",
        "p_category_id"
      ],
      "key": "parent",
      "used_key_parts": [
        "parent_id"
      ],
      "key_length": "3",
      "rows_examined_per_scan": 18,
      "rows_produced_per_join": 18,
      "filtered": "100.00",
      "index_condition": "(`jmj_test`.`jmj_categories`.`parent_id` in (67,68,82))",
      "cost_info": {
        "read_cost": "24.61",
        "eval_cost": "3.60",
        "prefix_cost": "28.21",
        "data_read_per_join": "69K"
      },
      "used_columns": [
        "category_id",
        "parent_id"
      ],
      "attached_condition": "(`jmj_test`.`jmj_categories`.`category_id` <> 1)"
    }
  }
}

Result

category_id	parent_id
69	67
70	67
71	67
72	67
73	67
74	67
75	67
76	67
77	67
83	68
84	68
85	68
86	68
104	82
106	82
105	82
107	82
108	82

category_id	parent_id
69	67
70	67
71	67
72	67
73	67
74	67
75	67
76	67
77	67
83	68
84	68
85	68
86	68
104	82
106	82
105	82
107	82
108	82

category_id	parent_id
69	67
70	67
71	67
72	67
73	67
74	67
75	67
76	67
77	67
83	68
84	68
85	68
86	68
104	82
106	82
105	82
107	82
108	82

category_id	parent_id
69	67
70	67
71	67
72	67
73	67
74	67
75	67
76	67
77	67
83	68
84	68
85	68
86	68
104	82
106	82
105	82
107	82
108	82