[...]
mysql> create table test(id int(11) not null, parent_id int(11) not null, primary key(id), key fk_parent (parent_id), constraint fk_parent foreign key(parent_id) references test(id) on delete cascade) engine=InnoDB;
Query OK, 0 rows affected (0.01 sec)

mysql> insert into test values (1,1), (2,1), (3,3);
Query OK, 3 rows affected (0.00 sec)
Records: 3  Duplicates: 0  Warnings: 0

mysql> delete from test where id=3;
Query OK, 1 row affected (0.00 sec)

mysql> delete from test where id=1;
Query OK, 1 row affected (0.01 sec)

mysql> select * from test;
Empty set (0.00 sec)

mysql> drop table test;
Query OK, 0 rows affected (0.01 sec)

mysql> Bye