For FIR, parallel typically refers to performing all (or more*) operations per sample per cycle. Sequential refers to doing one (or fewer**) operations per cycle.
There is a design method where filters are computed in parallel and then summed. This might be called "parallel" as well, but the argument would be "parallel" vs "series".
* eg, if you get 8 samples per cycle, you might do 8*N multiply/add per cycle. (eg, 4Gsps as 8 samples per 500MHz cycle)
** eg, if a multiply is a multi-cycle operation.