The optimization process and recommendations:

1. Avoid Calling Functions With Indexed Columns (query line: 21): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. For example, if the column `sysdate` is indexed, the index won’t be used as it’s wrapped with the function `to_date`. If you can’t find an alternative condition that won’t use a function call, a possible solution is to store the required value in a new indexed column.
2. Avoid Calling Functions With Indexed Columns (query line: 30): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. For example, if the column `sysdate` is indexed, the index won’t be used as it’s wrapped with the function `to_date`. If you can’t find an alternative condition that won’t use a function call, a possible solution is to store the required value in a new indexed column.
3. Avoid Calling Functions With Indexed Columns (query line: 39): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. For example, if the column `sysdate` is indexed, the index won’t be used as it’s wrapped with the function `to_date`. If you can’t find an alternative condition that won’t use a function call, a possible solution is to store the required value in a new indexed column.
4. Avoid Selecting Unnecessary Columns (query line: 2): Avoid selecting all columns with the '*' wildcard, unless you intend to use them all. Selecting redundant columns may result in unnecessary performance degradation.
5. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
6. Replace In Subquery With Correlated Exists (modified query below): In many cases, an EXISTS subquery with a correlated condition will perform better than a non correlated IN subquery.

Optimal indexes for this query:

CREATE INDEX tstapel_idx_fpartnrext ON tstapel (fpartnrext);

The optimized query:

SELECT
        * 
    FROM
        zdadba.tpartner partner 
    WHERE
        EXISTS (
            SELECT
                DISTINCT 1 
            FROM
                zdadba.tstapel stapel 
            WHERE
                (
                    stapel.lagernd = 1 
                    AND stapel.auftrag_id IS NULL 
                    AND (
                        (
                            CAST(stapel.version AS date) <= to_date('30.06.' || to_char(EXTRACT(YEAR 
                        FROM
                            sysdate)),
                            'DD.MM.YYYY')) 
                            AND (to_date(stapel.sysdate) >= to_date('30.06.' || to_char(EXTRACT(YEAR 
                        FROM
                            sysdate)),
                            'DD.MM.YYYY')) 
                            AND (partner.versandart = 'Halbjaehrlich')) 
                            OR ((CAST(stapel.version AS date) <= to_date('01.01.' || to_char(EXTRACT(YEAR 
                        FROM
                            sysdate)),
                            'DD.MM.YYYY')) 
                            AND (to_date(stapel.sysdate) <= to_date('30.06.' || to_char(EXTRACT(YEAR 
                        FROM
                            sysdate)),
                            'DD.MM.YYYY'))) 
                            AND partner.versandart = 'Halbjaehrlich' 
                            OR (((CAST(stapel.version AS date) <= to_date('01.01.' || to_char(EXTRACT(YEAR 
                        FROM
                            sysdate)),
                            'DD.MM.YYYY')) 
                            AND (to_date(stapel.sysdate) >= to_date('01.01.' || to_char(EXTRACT(YEAR 
                        FROM
                            sysdate)),
                            'DD.MM.YYYY'))) 
                            AND partner.versandart = 'Jaehrlich')) 
                            AND (partner.partnrext = stapel.fpartnrext)
                    )