Does there exist a problem that is hard to do in parallel?
Linear programming is P-complete with respect to either logspace or NC reductions (I'm not sure which).  As such, it's unlikely to be efficiently parallelizable because, for either kind of reductions, that would imply that P$\,=\,$NC. It's generally believed that P$\,\neq\,$NC.

There's more information on Wikipedia's P-completeness page.