SQL Server : Memory pressure,Spiked user connections and CPU usage
The threshold for parallelism should be higher than 20 for very large database. 

Memory pressure can be avoided by tuning query/index, rebuilding fragmented index, enabling TF 2371 at database level to fire auto update statistics frequently.

As a DBA you can add multiple file group to your very large database, each containing separate mdf file (located on different disk drive) and move larger table on newly added file group to get higher disk IO.