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We consider the $\varphi_{1,3}$ off-critical perturbation ${\cal M}(m,m';t)$of the general non-unitary minimal models where $2\le m\le m'$ and $m, m'$ arecoprime and $t$ measures the departure from criticality corresponding to the$\varphi_{1,3}$ integrable perturbation. We view these models as the continuumscaling limit in the ferromagnetic Regime III of the Forrester-BaxterRestricted Solid-On-Solid (RSOS) models on the square lattice. We also considerthe RSOS models in the antiferromagnetic Regime II related in the continuumscaling limit to $\mathbb{Z}_n$ parfermions with $n=m'-2$. Using an ellipticYang-Baxter algebra of planar tiles encoding the allowed face configurations,we obtain the Hamiltonians of the associated quantum chains defined as thelogarithmic derivative of the transfer matrices with periodic boundaryconditions. The transfer matrices and Hamiltonians act on a vector space ofpaths on the $A_{m'-1}$ Dynkin diagram whose dimension is counted bygeneralized Fibonacci numbers.

RSOS Quantum Chains Associated with Off-Critical Minimal Models and $\mathbb{Z}_n$ Parafermions