"mysql_num_row() so run this like a error check, return a error if 0 rows?"
mysql_num_rows() is not an error check, it just returns the number of records that your query returned.
If you run a query that finds no records, this function will return zero. Wether that is an error depends on your application.
"But if I put a part# 12-34, I get this Warning"
Most likely because there are no records where the partnumber is '12-34'.
Again, wether that is an error is up to you.