  In this paper, we demonstrate that not only the heat kernel techniques are
useful for computation of the parity anomaly, but also the parity anomaly turns
out to be a powerful mean in studying the heat kernel. We show that the
gravitational parity anomaly on 4D manifolds with boundaries can be calculated
using the general structure of the heat kernel coefficient $a_5$ for mixed
boundary conditions, keeping all the weights of various geometric invariants as
unknown numbers. The symmetry properties of the $\eta$-invariant allow to fix
all the relevant unknowns. As a byproduct of this calculation we get an
efficient and independent crosscheck (and confirmation) of the correction of
the general structure of $a_5$ for mixed boundary conditions, previously
suggested in Ref. [59].
