The documentation of read_sql_query says the following:

params : list, tuple or mapping, optional, default: None

    List of parameters to pass to execute method. The syntax used to pass parameters is database driver dependent. Check your database driver documentation for which of the five syntax styles, described in PEP 249’s paramstyle, is supported. Eg. for psycopg2, uses %(name)s so use params={‘name’ : ‘value’}.

Since you use the psycopg2 driver the parameters should be noted as @JonSG has mentioned. It should be:

select *
FROM public.bq_results br 
WHERE cast("eventDate" as date) between 
  TO_DATE(%(test_start_date)s, 'YYYYMMDD') AND TO_DATE(%(test_end_date)s, 'YYYYMMDD')
limit 10000

Hope this works.