  The celebrated phenomenon of quantum Hall effect has recently been
generalized from transport of conserved charges to that of other approximately
conserved state variables, including spin and valley, which are characterized
by spin- or valley-polarized boundary states with different chiralities. Here,
we report a new class of quantum Hall effect in ABA-stacked graphene trilayers
(TLG), the quantum parity Hall (QPH) effect, in which boundary channels are
distinguished by even or odd parity under the systems mirror reflection
symmetry. At the charge neutrality point and a small perpendicular magnetic
field $B_{\perp}$, the longitudinal conductance $\sigma_{xx}$ is first
quantized to $4e^2/h$, establishing the presence of four edge channels. As
$B_{\perp}$ increases, $\sigma_{xx}$ first decreases to $2e^2/h$, indicating
spin-polarized counter-propagating edge states, and then to approximately $0$.
These behaviors arise from level crossings between even and odd parity bulk
Landau levels, driven by exchange interactions with the underlying Fermi sea,
which favor an ordinary insulator ground state in the strong $B_{\perp}$ limit,
and a spin-polarized state at intermediate fields. The transitions between
spin-polarized and unpolarized states can be tuned by varying Zeeman energy.
Our findings demonstrate a topological phase that is protected by a
gate-controllable symmetry and sensitive to Coulomb interactions.
