  We theoretically explore the emergence of strong zero modes in a two-site
chain consisting of two quantum dots coupled due to a central dot that mediates
electron hopping and singlet superconducting pairing. In the presence of
time-reversal symmetry, the on-site Coulomb interaction leads to a three-fold
ground-state degeneracy when tuning the system to a sweet spot as a function of
the inter-dot couplings. This degeneracy is protected against changes of the
dot energies in the same way as "poor man's'' Majorana bound states in short
Kitaev chains. In the limit of strong interactions, this protection is maximal
and the entire spectrum becomes triply degenerate, indicating the emergence of
a ''poor man's'' version of a strong zero mode. We explain the degeneracy and
protection by constructing corresponding Majorana Kramers-pair operators and
$\mathbb{Z}_3$-parafermion operators. The strong zero modes share many
properties of Majorana bound states in short Kitaev chains, including the
stability of zero-bias peaks in the conductance and the behavior upon coupling
to an additional quantum dot. However, they can be distinguished through
finite-bias spectroscopy and the exhibit a different behavior when scaling to
longer chains.
