I don't remember precisely what are the terms in magnetohydrodynamics energy momentum tensor  but they should be invariant under parity transformation since E (P) is parity odd and B(M) is parity even. As I know,  in magnetohydrodynamics energy-momentum tensor there are no coupling between E(P) and B(M) but this coupling occurs only in current term indirectly by non-conservation law of anomalous current. Time reversal symmetry in anomalous phenomena is more important than parity symmetry. This is the time reversal symmetry in which one can deduce the existence of anomalous transport phenomena such as those in Son's paper arXiv:0906.5044 [hep-th]. 

There is actually a term that combines E and B, however it seems to be usually neglected. This term is related to the local Poynting vector which allows electromagnetic energy to move between parcels. All other terms, as far as I know, are even. If you include this term then the overall tensor is neither even nor odd. In addition this term maximally breaks boost-invariance.

It depends on how many spacial dimensions are being flipped. There is a choice. If you are flipping two dimensions perpendicular to the force and applying the relativistic (differential) form of Maxwell Equations then the force vector is invariant, although all the electric and magnetic fields reverse. This explains why AC power can be applied to the magneto hydrodynamic device

With other choices of flipped dimensions you get a variety of results. Generally flipping one or three dimensions  makes a sign reversal in the force vector which is not an invariant response and not the usual requirement of a system.

I need to think more on your answer, however I agree with your final point. It needs to be a requirement. However this makes the the energy-momentum tensor not diagonal in the local rest frame. This is again not a requirement but seems to take the fluid out of local equilibrium.

Aren't pressure and energy density odd themselves?