  This note provides some new perspectives and calculations regarding an
interesting known family of minimal surfaces in $\mathbb{H}^2 \times
\mathbb{R}$. The surfaces in this family are the catenoids, parabolic catenoids
and tall rectangles. Each is foliated by either circles, horocycles or circular
arcs in horizontal copies of $\mathbb{H}^2$. All of these surfaces are
well-known, but the emphasis here is on their unifying features and the fact
that they lie in a single continuous family. We also initiate a study of the
Jacobi operator on the parabolic catenoid, and compute the Jacobi fields
associated to deformations to either of the two other types of surfaces in this
family.
