How to specify the parent query field from within a subquery in mySQL?

Give the tables unique names:

$query = "SELECT a.id, (SELECT COUNT(1) FROM post_table b WHERE parent_id = a.id) as num_children FROM post_table a WHERE a.parent_id = 0";

How about:

$query = "SELECT p1.id, 
                 (SELECT COUNT(1) 
                    FROM post_table p2 
                   WHERE p2.parent_id = p1.id) as num_children
            FROM post_table p1
           WHERE p1.parent_id = 0";

or if you put an alias on the p1.id, you might say:

$query = "SELECT p1.id as p1_id, 
                 (SELECT COUNT(1) 
                    FROM post_table p2 
                   WHERE p2.parent_id = p1.id) as num_children
            FROM post_table p1
           WHERE p1.parent_id = 0";

You could try something like this

SELECT  pt.id,
        CountTable.Cnt
FROM    post_table pt LEFT JOIN
        (
            SELECT  parent_id,
                    COUNT(1) Cnt
            FROM    post_table
            WHERE   parent_id <> 0
            GROUP BY parent_id
        ) CountTable ON pt.id = CountTable.parent_id
WHERE   pt.parent_id = 0

To get back to your example, use the alias of the main table in the sub select

SELECT  pt.id,
        (SELECT COUNT(1) FROM post_table WHERE parent_id = pt.id) 
FROM    post_table pt
WHERE   pt.parent_id = 0