Recursive CTE to find Total for all children
This recursive CTE (SQL Fiddle) should work with your sample:

WITH cte(ParentID) AS(
    SELECT ParentID FROM @Instances WHERE [Part] = 'Rivet'
    UNION ALL
    SELECT i.ParentID FROM cte c
    INNER JOIN @Instances i ON c.ParentID = i.InstanceID
    WHERE i.ParentID &gt; 0
)
SELECT ParentID, count(*) 
FROM cte
GROUP BY ParentID
ORDER BY ParentID
;


Output

ParentID    Count
1           6
2           3
3           2
4           1


Note: You mentioned in comments that the question only contains a simplified sample table and real data have proper indexes and handle duplicates and data adequately.  

Data used (SQL Fiddle):

DECLARE @Instances TABLE( 
    [InstanceID] int NOT NULL
    , [Part] NVARCHAR (50) NOT NULL
    , [ParentID] int NOT NULL
);

INSERT INTO @Instances([InstanceID], [Part], [ParentID])
VALUES 
    (1, 'Assembly', 0)
    , (50, 'Rivet', 1)
    , (50, 'Rivet', 1)
    , (2, 'SubAssembly', 1)
    , (50, 'Rivet', 2)
    , (51, 'Bolt', 2)
    , (51, 'Bolt', 2)
    , (3, 'SubSubAssembly', 2)
    , (50, 'Rivet', 3)
    , (50, 'Rivet', 3)
    , (4, 'SubAssembly2', 1)
    , (50, 'Rivet', 4)
    , (51, 'Bolt', 4)
;