When in doubt, tinker. Tinker with what you are allowed to change (which doesn't sound like much). Can you set MAXDOP to 4? MAXDOP 0 can be a killer as it'll bulk up on threads for a query on the assumption nothing else is going to run. A few milliseconds later when 100 similar queries have been executed, VOILA! Thousands of extra threads. For my dollar, even with hyperthreading MAXDOP should be set no higher than the number of physical cores - half that in AMD environments or busy Intel environments.

Usually things are disk bound anyway. Sure would be nice to install a few SSDs and use them for TEMPDB - that can make a huge difference everywhere.

Oh - and bad statistics. That could indirectly cause a lot of parallelism and extra threads.