  We give a formula for the Carath\'eodory distance on the Neil parabola, the
variety ${z^2=w^3}$ restricted to the bidisk; thus making it the first variety
with a singularity to have its Carath\'eodory distance explicitly computed. In
addition, we relate this to a mixed Carath\'eodory-Pick interpolation problem
for which known interpolation theorems do not apply. Finally, we prove a
bounded holomorphic function extension result from the Neil parabola to the
bidisk.
