  Incompatibility of quantum devices is one of the cornerstones of quantum
theory, and the incompatibility of quantum measurements and channels has been
linked to quantum advantage in certain information theoretic tasks. In this
work, we focus on the less well-explored question of the incompatibility of
quantum instruments, that is, devices that describe the measurement process in
its entirety, accounting for both the classical measurement outcome and the
quantum post-measurement state. In particular, we focus on the recently
introduced notion of parallel compatibility of instruments, which has been
argued to be a natural notion of instrument compatibility. We introduce --
similarly to the case of measurements and channels -- the incompatibility
robustness of quantum instruments and derive universal bounds on it. We then
prove that post-processing of quantum instruments is a free operation for
parallel compatibility. Last, we provide families of instruments for which our
bounds are tight, and families of compatible indecomposable instruments.
