  We consider the newly proposed Bahamonde-Dialektopoulos-Levi Said (BDLS)
theory, that is the Horndeski analog in the teleparallel framework and thus
contains a non-minimally coupled scalar field, including higher order
derivatives, that leads however to second order field equations both for the
tetrad and the scalar field. This theory was mostly constructed to revive those
models that were severely constrained in the scalar-tensor version of the
theory from the GW170817, but includes also much richer phenomenology because
of the nature of the torsion tensor. For this theory we determine the
parametrized post-Newtonian limit, calculate the full set of post-Newtonian
parameters and highlight some special cases.
