SELECT 
  p.product_id, 
  ifnull(
    count(dp.post_id), 
    0
  ) as discussion_amount_posts 
FROM 
  cscart_discussion 
  INNER JOIN cscart_products as p ON (
    cscart_discussion.object_id = p.product_id
  ) 
  INNER JOIN cscart_discussion_posts as dp ON (
    cscart_discussion.thread_id = dp.thread_id 
    AND cscart_discussion.object_type = 'P' 
    AND cscart_discussion.company_id = 1
  ) 
WHERE 
  dp.status = 'A' 
  and p.product_id in (125, 120, 117, 113, 68) 
GROUP BY 
  p.product_id

Query time 0.00031

JSON explain

{
  "query_block": {
    "select_id": 1,
    "nested_loop": [
      {
        "table": {
          "table_name": "p",
          "access_type": "range",
          "possible_keys": ["PRIMARY"],
          "key": "PRIMARY",
          "key_length": "3",
          "used_key_parts": ["product_id"],
          "rows": 5,
          "filtered": 100,
          "attached_condition": "p.product_id in (125,120,117,113,68)",
          "using_index": true
        }
      },
      {
        "table": {
          "table_name": "cscart_discussion",
          "access_type": "eq_ref",
          "possible_keys": ["PRIMARY", "object_id", "idx_company_id"],
          "key": "object_id",
          "key_length": "10",
          "used_key_parts": ["object_id", "object_type", "company_id"],
          "ref": ["cscart_4183_25.p.product_id", "const", "const"],
          "rows": 1,
          "filtered": 100,
          "index_condition": "cscart_discussion.object_type = 'P'"
        }
      },
      {
        "table": {
          "table_name": "dp",
          "access_type": "ref",
          "possible_keys": ["thread_id", "thread_id_2"],
          "key": "thread_id_2",
          "key_length": "6",
          "used_key_parts": ["thread_id", "status"],
          "ref": ["cscart_4183_25.cscart_discussion.thread_id", "const"],
          "rows": 1,
          "filtered": 100,
          "index_condition": "dp.`status` = 'A'"
        }
      }
    ]
  }
}