  We study the parametrized post-Newtonian (PPN) limit of
higher-derivative-torsion Modified Teleparallel Gravity. We start from the
covariant formulation of modified Teleparallel Gravity by restoring the spin
connection of the theory. Then, we perform the post-Newtonian expansion of the
tetrad field around the Minkowski background and find the perturbed field
equations. We compute the PPN metric for the higher-order Teleparallel Gravity
theories which allows us to show that at the post-Newtonian limit this more
general class of theories are fully conservative and indistinguishable from
General Relativity . In this way, we extend the results that were already found
for $F(T)$ gravity in previous works. Furthermore, our calculations reveal the
importance of considering a second post-Newtonian (2PN) order approximation or
a parametrized post-Newtonian cosmology (PPNC) framework where additional
perturbative modes coming from general modifications of Teleparallel Gravity
could lead to new observable imprints.
