  We study a model of nonintersecting Brownian bridges on an interval with
either absorbing or reflecting walls at the boundaries, focusing on the point
in space-time at which the particles meet the wall. These processes are
determinantal, and in different scaling limits when the particles approach the
reflecting (resp. absorbing) walls we obtain hard-edge limiting kernels which
are the even (resp. odd) parts of the Pearcey and tacnode kernels. We also show
that in the single time case, our hard-edge tacnode kernels are equivalent to
the ones studied by Delvaux [16], defined in terms of a $4\times 4$ Lax pair
for the inhomogeneous Painlev\'{e} II equation (PII). As a technical ingredient
in the proof, we construct a Schlesinger transform for the $4 \times 4$ Lax
pair in [16] which preserves the Hastings--McLeod solutions to PII.
