  The development of non-Hermitian topological band theory has led to
observations of novel topological phenomena in effectively classical, driven
and dissipative systems. However, for open quantum many-body systems, the
absence of a ground state presents a challenge to define robust signatures of
non-Hermitian topology. We show that such a signature is provided by crossings
in the time evolution of the entanglement spectrum. These crossings occur in
quenches from the trivial to the topological phase of a driven-dissipative
Kitaev chain that is described by a Markovian quantum master equation in
Lindblad form. At the topological transition, which can be crossed either by
changing parameters of the Hamiltonian of the system or by increasing the
strength of dissipation, the time scale at which the first entanglement
spectrum crossing occurs diverges with a dynamical critical exponent of
$\epsilon = 1/2$. We corroborate these numerical findings with an exact
analytical solution of the quench dynamics for a spectrally flat postquench
Liouvillian. This exact solution suggests an interpretation of the topological
quench dynamics as a fermion parity pump. Our work thus reveals signatures of
non-Hermitian topology which are unique to quantum many-body systems and cannot
be emulated in classical simulators of non-Hermitian wave physics.
