  Unconventional lattice fermions with high degeneracies beyond Weyl and Dirac
fermions have attracted intensive attention in recent years. In this paper,
attention is drawn to the pseudospin-1 Maxwell fermions and the $(2+1)$
dimensional parity anomaly, which goes beyond the scope of "fermion doubling
theorem". We have derived the Hall conductivity of a single Maxwell fermion,
and showcased each Maxwell fermion contributes a quantized Hall conductance
$e^{2}/h$. We observe that the parity is spontaneously broken in the effective
theory of lattice Maxwell fermions interacting with an (auxiliary) $U(1)$ gauge
field, leading to an effective anomaly-induced Chern-Simons theory. An
interesting observation from the parity anomaly is that the lattice Maxwell
fermions beyond the "fermion doubling theorem", so there exists single Maxwell
fermion in a lattice model. In addition, our work indicates the quantum anomaly
in odd-dimensional spinor space.
