  Quantum parity conservation is verified at all orders in perturbation theory
for a massless parity-even $U(1)\times U(1)$ planar quantum electrodynamics
(QED$_3$) model. The presence of two massless fermions requires the
Lowenstein-Zimmermann (LZ) subtraction scheme, in the framework of the
Bogoliubov-Parasiuk-Hepp-Zimmermann-Lowenstein (BPHZL) renormalization method,
in order to subtract the infrared divergences induced by the ultraviolet
subtractions at 1- and 2-loops, however thanks to the superrenormalizability of
the model the ultraviolet divergences are bounded up to 2-loops. Finally, it is
proved that the BPHZL renormalization method preserves parity for the model
taken into consideration, contrary to what happens to the ordinary massless
parity-even $U(1)$ QED$_3$.
