  We study entanglement dynamics in hybrid $\mathbb{Z}_2$-symmetric quantum
automaton circuits subject to local composite measurements. We show that there
exists an entanglement phase transition from a volume law phase to a critical
phase by varying the measurement rate $p$. By analyzing the underlying
classical bit string dynamics, we demonstrate that the critical point belongs
to parity-conserving universality class. We further show that the critical
phase with $p>p_c$ is related to the diffusion-annihilation process and is
protected by the $\mathbb{Z}_2$-symmetric measurement. We give an
interpretation of the entanglement entropy in terms of a two-species particle
model and identify the coefficient in front of the critical logarithmic
entanglement scaling as the local persistent coefficient. The critical behavior
observed at $p\geq p_c$ and the associated dynamical exponents are also
confirmed in the purification dynamics.
