In SQL Server, is parallelism per operator, or something else?
He says that in a parallel plan, every operator gets DOP threads.


No. This is at best misleading, but closer to being simply wrong.

In a serial plan, every operator 'gets' one thread, but that thread is the same thread for all operators. The principle is similar for parallel plans.

Each parallel operator is run by DOP threads, but those threads are not exclusive to a particular operator, they are shared among operators within the same parallel branch.

Branch boundaries are delimited by Parallelism operators (Demand, Repartition, and Gather Streams). The diagram below shows a parallel plan with three branches:


Reproduced from the article referenced at the end of this answer




  So if you have MAXDOP 8 and your query has 4 parallel operators it'll use 32 threads at once.


No. You can't just multiply DOP by the number of operators to get the number of threads. The number of threads reserved for parallel branches is the number of parallel branches (not operators) multiplied by DOP.

The number of threads that can be active at the same time for a single parallel query is limited to DOP in SQL Server 2005 and later. SQL Server achieves this by allocating threads to DOP schedulers.




  I also read that it might just be 8 for the whole query, which seems like too few.


At DOP = 8 this would be correct for a plan with a single parallel branch. There can be multiple parallel plan branches in a parallel plan. For a plan with n parallel branches, the thread reservation for parallel workers is n * DOP.

See Parallel Execution Plans – Branches and Threads by Paul White.

Note: The number of branches reported in execution plans is normally the number of branches that may possibly execute concurrently (due to blocking operators, threads in one branch may sometimes be safely recycled for a later branch).