Select from one table where not in another

You can LEFT JOIN the two tables. If there is no corresponding row in the second table, the values will be NULL.

SELECT id FROM partmaster LEFT JOIN product_details ON (...) WHERE product_details.part_num IS NULL

Expanding on Sjoerd's anti-join, you can also use the easy to understand SELECT WHERE X NOT IN (SELECT) pattern.

SELECT pm.id FROM r2r.partmaster pm
WHERE pm.id NOT IN (SELECT pd.part_num FROM wpsapi4.product_details pd)

Note that you only need to use ` backticks on reserved words, names with spaces and such, not with normal column names.

On MySQL 5+ this kind of query runs pretty fast.
On MySQL 3/4 it's slow.

Make sure you have indexes on the fields in question
You need to have an index on pm.id, pd.part_num.