We study three-dimensional noncompact QED with a single two-component massless fermion and two infinitely massive regulator fermions of half the charge using the lattice overlap formalism. The parity anomaly is expected to cancel exactly between the massless and regulator fermions in the continuum, but this cancellation is inexact on a lattice akin to lattice chiral gauge theories. We show nonperturbatively that parity-breaking terms vanish in the continuum limit at any finite volume. We present numerical evidence that the resulting parity-invariant theory spontaneously breaks parity in the infinite volume limit.