  The future space-based gravitational wave detector LISA will be able to
measure parameters of coalescing massive black hole binaries, often to
extremely high accuracy. Previous work has demonstrated that the black hole
spins can have a strong impact on the accuracy of parameter measurement.
Relativistic spin-induced precession modulates the waveform in a manner which
can break degeneracies between parameters, in principle significantly improving
how well they are measured. Recent studies have indicated, however, that spin
precession may be weak for an important subset of astrophysical binary black
holes: those in which the spins are aligned due to interactions with gas. In
this paper, we examine how well a binary's parameters can be measured when its
spins are partially aligned and compare results using waveforms that include
higher post-Newtonian harmonics to those that are truncated at leading
quadrupole order. We find that the weakened precession can substantially
degrade parameter estimation. This degradation is particularly devastating for
the extrinsic parameters sky position and distance. Absent higher harmonics,
LISA typically localizes the sky position of a nearly aligned binary a factor
of $\sim 6$ less accurately than for one in which the spin orientations are
random. Our knowledge of a source's sky position will thus be worst for the
gas-rich systems which are most likely to produce electromagnetic counterparts.
Fortunately, higher harmonics of the waveform can make up for this degradation.
By including harmonics beyond the quadrupole in our waveform model, we find
that the accuracy with which most of the binary's parameters are measured can
be substantially improved. In some cases, parameters can be measured as well in
partially aligned binaries as they can be when the binary spins are random.
