以下、Connector/Python ドライバを使用している前提で回答させていただきます。

MySQL のマニュアルの INSERT ... ON DUPLICATE KEY UPDATE 構文のページに、以下のように説明があります。
https://dev.mysql.com/doc/refman/5.6/ja/insert-on-duplicate.html

ON DUPLICATE KEY UPDATE を使用した場合、行ごとの影響を受けた行の値は、その行が新しい行として挿入された場合は 1、既存の行が更新された場合は 2、既存の行がその現在の値に設定された場合は 0 です。

私の手元の環境(※)で試した限りでは、この動作は MySQLCursor.rowcount プロパティでも同様です。
※ Mac OSX 10.5, Python 2.7.10, MySQL Connector/Python 2.2

そのため、以下のように 1行ずつ INSERT ... ON DUPLICATE KEY UPDATE し、
rowcount プロパティの値を確認すれば

更新件数、追加件数、スキップ件数

を取得できます。

■テーブル定義

mysql> CREATE TABLE test (
    ->   id INT PRIMARY KEY AUTO_INCREMENT,
    ->   value VARCHAR(4)
    -> );
Query OK, 0 rows affected (0.03 sec)

mysql> INSERT INTO test VALUES
    -> (null, 'a'),
    -> (null, 'b'),
    -> (null, 'c');
Query OK, 3 rows affected (0.00 sec)
Records: 3  Duplicates: 0  Warnings: 0

mysql> select * from test;
+----+-------+
| id | value |
+----+-------+
|  1 | a     |
|  2 | b     |
|  3 | c     |
+----+-------+
3 rows in set (0.00 sec)

■ソースコード

import mysql.connector

cnx = mysql.connector.connect(user='XXXX', password='XXXX',
                              host='XXXX',
                              database='XXXX')

cursor = cnx.cursor()

skipcount = 0
insertcount = 0
updatecount = 0

sql = ("INSERT INTO test VALUES (%s, %s)"
        "ON DUPLICATE KEY UPDATE value = VALUES(value)")
parameters = [
    (1, 'a'), # skip
    (2, 'd'), # update
    (3, 'e'), # update
    (4, 'f'), # insert
    (5, 'g'), # insert
    (6, 'h'), # insert
]

for parameter in parameters:
    cursor.execute(sql, parameter)

    rowcount = cursor.rowcount
    if rowcount == 0:
        skipcount += 1
    elif rowcount == 1:
        insertcount += 1
    elif rowcount == 2:
        updatecount += 1
    else:
        print "Something is wrong!"

cnx.commit()
cnx.close()

print skipcount
print insertcount
print updatecount

■実行結果

$ python 72041.py 
1
3
2

mysql> select * from test;
+----+-------+
| id | value |
+----+-------+
|  1 | a     |
|  2 | d     |
|  3 | e     |
|  4 | f     |
|  5 | g     |
|  6 | h     |
+----+-------+
6 rows in set (0.00 sec)

ちなみに、yambejp様の回答の通り、INSERT ... ON DUPLICATE KEY UPDATE 構文に IGNORE キーワードを付ける必要はありません。
付けてもシンタックスエラーにはなりませんが、SQL文の実行結果は IGNORE キーワードを付けなかった場合と変わりありません。

mysql> SELECT * FROM test;
+----+-------+
| id | value |
+----+-------+
|  1 | a     |
|  2 | d     |
|  3 | e     |
|  4 | f     |
|  5 | g     |
|  6 | h     |
+----+-------+
6 rows in set (0.01 sec)

mysql> BEGIN;
Query OK, 0 rows affected (0.00 sec)

mysql> INSERT INTO test VALUES (1, 'a'), (2, 'i'), (7, 'j') ON DUPLICATE KEY UPDATE value = VALUES(value);
Query OK, 3 rows affected (0.00 sec)
Records: 3  Duplicates: 1  Warnings: 0

mysql> SELECT * FROM test;
+----+-------+
| id | value |
+----+-------+
|  1 | a     |
|  2 | i     |
|  3 | e     |
|  4 | f     |
|  5 | g     |
|  6 | h     |
|  7 | j     |
+----+-------+
7 rows in set (0.00 sec)

mysql> ROLLBACK;
Query OK, 0 rows affected (0.00 sec)

mysql> BEGIN;
Query OK, 0 rows affected (0.00 sec)

mysql> INSERT IGNORE INTO test VALUES (1, 'a'), (2, 'i'), (7, 'j') ON DUPLICATE KEY UPDATE value = VALUES(value);
Query OK, 3 rows affected (0.00 sec)
Records: 3  Duplicates: 1  Warnings: 0

mysql> SELECT * FROM test;
+----+-------+
| id | value |
+----+-------+
|  1 | a     |
|  2 | i     |
|  3 | e     |
|  4 | f     |
|  5 | g     |
|  6 | h     |
|  7 | j     |
+----+-------+
7 rows in set (0.00 sec)

mysql> ROLLBACK;
Query OK, 0 rows affected (0.00 sec)

• おそらくバルクインサートしたいと思われるので、こんなやり方もあるよ的な
• 日時のカラム(created_at,updated_at)がある前提です
• 件数が多い場合は created_at,updated_at で複合index貼っておくよいです

 プログラム側で日付を渡す場合

• skipは取得できないです、更新としてカウントされます

mysql> show create table t1;
+-------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| Table | Create Table                                                                                                                                                                                                   |
+-------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| t1    | CREATE TABLE `t1` (
  `id` int(11) NOT NULL,
  `name` varchar(255) DEFAULT NULL,
  `created_at` datetime NOT NULL,
  `updated_at` datetime NOT NULL,
  PRIMARY KEY (`id`)
) ENGINE=InnoDB DEFAULT CHARSET=utf8 |
+-------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
1 row in set (0.00 sec)

mysql> insert into t1 (id,name,created_at,updated_at)  values (1,'a','2017-04-01','2017-04-01'),(2,'b','2017-04-01','2017-04-01'),(3,'c','2017-04-01','2017-04-01');
Query OK, 3 rows affected (0.00 sec)
Records: 3  Duplicates: 0  Warnings: 0

mysql> select * from t1;
+----+------+---------------------+---------------------+
| id | name | created_at          | updated_at          |
+----+------+---------------------+---------------------+
|  1 | a    | 2017-04-01 00:00:00 | 2017-04-01 00:00:00 |
|  2 | b    | 2017-04-01 00:00:00 | 2017-04-01 00:00:00 |
|  3 | c    | 2017-04-01 00:00:00 | 2017-04-01 00:00:00 |
+----+------+---------------------+---------------------+
3 rows in set (0.00 sec)

mysql> insert into t1 (id,name,created_at,updated_at)  values (1,'a','2017-04-02','2017-04-02'),(2,'bb','2017-04-02','2017-04-02'),(3,'cc','2017-04-02','2017-04-02'),(4,'d','2017-04-02','2017-04-02') on duplicate key update name=values(name),updated_at=values(updated_at);
Query OK, 7 rows affected (0.00 sec)
Records: 4  Duplicates: 3  Warnings: 0

mysql> select * from t1;
+----+------+---------------------+---------------------+
| id | name | created_at          | updated_at          |
+----+------+---------------------+---------------------+
|  1 | a    | 2017-04-01 00:00:00 | 2017-04-02 00:00:00 |
|  2 | bb   | 2017-04-01 00:00:00 | 2017-04-02 00:00:00 |
|  3 | cc   | 2017-04-01 00:00:00 | 2017-04-02 00:00:00 |
|  4 | d    | 2017-04-02 00:00:00 | 2017-04-02 00:00:00 |
+----+------+---------------------+---------------------+
4 rows in set (0.00 sec)

mysql> select (select count(*) from t1 where created_at=updated_at and updated_at='2017-04-02') as add_count, (select count(*) from t1 where created_at<updated_at and updated_at='2017-04-02') as update_count;
+-----------+--------------+
| add_count | update_count |
+-----------+--------------+
|         1 |            3 |
+-----------+--------------+
1 row in set (0.00 sec)

 日時を自動でセットするカラムがある場合

• 別途最新レコードを取得してupdated_atの値を取得する必要ありますが
• スキップ件数 = 対象件数 - add_count - update_count

mysql> show create table t2;
+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| Table | Create Table                                                                                                                                                                                                                                                                                   |
+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| t2    | CREATE TABLE `t2` (
  `id` int(11) NOT NULL,
  `name` varchar(255) DEFAULT NULL,
  `created_at` datetime NOT NULL DEFAULT CURRENT_TIMESTAMP,
  `updated_at` datetime NOT NULL DEFAULT CURRENT_TIMESTAMP ON UPDATE CURRENT_TIMESTAMP,
  PRIMARY KEY (`id`)
) ENGINE=InnoDB DEFAULT CHARSET=utf8 |
+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
1 row in set (0.00 sec)

mysql> insert into t2 (id,name)  values (1,'a'),(2,'b'),(3,'c');
Query OK, 3 rows affected (0.00 sec)
Records: 3  Duplicates: 0  Warnings: 0

mysql> select * from t2;
+----+------+---------------------+---------------------+
| id | name | created_at          | updated_at          |
+----+------+---------------------+---------------------+
|  1 | a    | 2017-04-11 07:50:19 | 2017-04-11 07:50:19 |
|  2 | b    | 2017-04-11 07:50:19 | 2017-04-11 07:50:19 |
|  3 | c    | 2017-04-11 07:50:19 | 2017-04-11 07:50:19 |
+----+------+---------------------+---------------------+
3 rows in set (0.00 sec)

mysql> insert into t2 (id,name)  values (1,'a'),(2,'bb'),(3,'cc'),(4,'d') on duplicate key update name=values(name);
Query OK, 5 rows affected (0.00 sec)
Records: 4  Duplicates: 2  Warnings: 0

mysql> select * from t2;
+----+------+---------------------+---------------------+
| id | name | created_at          | updated_at          |
+----+------+---------------------+---------------------+
|  1 | a    | 2017-04-11 07:50:19 | 2017-04-11 07:50:19 |
|  2 | bb   | 2017-04-11 07:50:19 | 2017-04-11 07:51:00 |
|  3 | cc   | 2017-04-11 07:50:19 | 2017-04-11 07:51:00 |
|  4 | d    | 2017-04-11 07:51:00 | 2017-04-11 07:51:00 |
+----+------+---------------------+---------------------+
4 rows in set (0.00 sec)

mysql> select (select count(*) from t2 where created_at=updated_at and updated_at='2017-04-11 07:51:00') as add_count, (select count(*) from t2 where created_at<updated_at and updated_at='2017-04-11 07:51:00') as update_count;
+-----------+--------------+
| add_count | update_count |
+-----------+--------------+
|         1 |            2 |
+-----------+--------------+
1 row in set (0.00 sec)