You can combine the two Selects in a SQL JOIN so that the limited set of rows from table1 is joined with only relevant details from Table2.
A QlikView alternative consists of first storing the list of relevant PART_ID values in a variable and then use that variable in each WHERE clause. For example, something like this (off the top of my head, some tuning may be required):
SQL SELECT "PART_ID"
FROM "DATABASE1" where PART_NUMBER IN ('84171490','84017929','9519027','9998266');