Number of Worker threads available in PostgreSQL
PostgreSQL does not use threads, it uses coordinated single-thread processes.  It will not use more than max_parallel_workers, system-wide, at one time.  If it would exceed that, the queries will not error out, they simply run with fewer worker processes than they "wanted".

If the number of connections you have trying to run queries at the same time is always greater than your number of CPUs, then you probably might as well turn parallelism off.  OLTP usually are not great candidates for parallel execution anyway.