SELECT 
  lnk.link_id, 
  lnk.usergroup_id, 
  lnk.status 
FROM 
  cscart_usergroup_links as lnk 
  INNER JOIN cscart_usergroups ON cscart_usergroups.usergroup_id = lnk.usergroup_id 
  AND cscart_usergroups.status != 'D' 
WHERE 
  lnk.user_id = 13 
  AND lnk.status IN ('A')

Query time 0.00028

JSON explain

{
  "query_block": {
    "select_id": 1,
    "nested_loop": [
      {
        "table": {
          "table_name": "lnk",
          "access_type": "ref",
          "possible_keys": ["user_id"],
          "key": "user_id",
          "key_length": "3",
          "used_key_parts": ["user_id"],
          "ref": ["const"],
          "rows": 1,
          "filtered": 100,
          "attached_condition": "lnk.`status` = 'A'"
        }
      },
      {
        "table": {
          "table_name": "cscart_usergroups",
          "access_type": "eq_ref",
          "possible_keys": ["PRIMARY", "c_status", "status"],
          "key": "PRIMARY",
          "key_length": "3",
          "used_key_parts": ["usergroup_id"],
          "ref": ["v8.lnk.usergroup_id"],
          "rows": 1,
          "filtered": 100,
          "attached_condition": "cscart_usergroups.`status` <> 'D'"
        }
      }
    ]
  }
}