SQL Server experiencing THREADPOOL waits while still having plenty of available workers
If you have the max degree of parallelism set to zero, which is the default, then queries that might benefit from parallelism will consume 32 threads.  With the number of workers set to 960, again the default for your setup, you could only run 30 simultaneous parallel queries.

I'd recommend setting max degree of parallelism to something sane, like 8.