MSSQL cast( [varcharColumn] to int) in SELECT gets executed before WHERE clause filters out bad values

You can't easily control the way SQL Server processes your query. You can figure out some of the why by deep diving into the execution plan, but understanding that is the least of your problems in this specific case I think. You can do a little with join hints, perhaps, but that's hacky to me and the behavior is still not guaranteed (especially as you move to new versions etc). Two workarounds you could try are:

;WITH c AS 
(
  SELECT varcharColumn, ParentID, TypeId
   FROM dbo.Child AS c
   WHERE c.TypeId = 2
   AND ISNUMERIC(varcharColumn) = 1 --*
)
SELECT CONVERT(INT, c.varcharColumn)
FROM dbo.Parent AS p
INNER JOIN c
ON c.ParentId = p.Id
WHERE p.TypeId = 13;

But I have heard of cases where even separating this out into a CTE could lead to the bad plan that led the convert to occur first. So it may be that you need to break it out even further:

SELECT varcharColumn, ParentID, TypeId
INTO #c
   FROM dbo.Child AS c
   WHERE c.TypeId = 2
   AND ISNUMERIC(varcharColumn) = 1; --*

SELECT CONVERT(INT, c.varcharColumn)
  FROM dbo.Parent AS p
  INNER JOIN #c AS c
  ON c.ParentId = p.Id
  WHERE p.TypeId = 13;

(I also talk about the CASE expression solution in this answer.)

If you are on SQL Server 2012, you can simply do this - now it doesn't matter if the convert is attempted before the filter, and you don't have to rely on the wonky ISNUMERIC() function.*

SELECT TRY_CONVERT(INT, c.varcharColumn)
  FROM dbo.Parent AS p
  INNER JOIN dbo.Child AS c
  ON c.ParentId = p.Id
  WHERE c.TypeId = 2
  AND p.TypeId = 13;

* Please note that IsNumeric is not perfect. I wrote this article several years ago to help deal with this.

First, this is not a "glaring design issue". SQL is a descriptive language of the output, not a procedural language that specifies how prcoessing is being done. There is no guarantee of the order of processing, in general, and this is an advantage. I might say there is a design issue, but it is around the general handling of exceptions in SQL statements.

According to SQL Server documentation (http://msdn.microsoft.com/en-us/library/ms181765.aspx), you can depend on the order of evauation for a CASE statement for scalar expressions. So, the following should work:

select (case when isnumeric(c.varcharColumn) = 1 then cast(c.varcharColumn as int) end)

Or, to get closer to an "int" expression:

select (case when isnumeric(c.varcharColumn) = 1 and c.varcharColumn not like '%.%' and c.varcharColumn not like '%e%'
             then cast(c.varcharColumn as int)
        end)

At least your code is doing an explicit CAST. This situation is much nastier when the casts are implicit (and there are hundreds of columns).