  In this paper, we consider the possible effect of the teleparallel gravity on
the production of the primordial black holes (PBH) and on the gravitational
waves (GWs). We investigate the relationship between the slow roll, the e-folds
number and the teleparallel parameters. We show that in the case of the
teleparallel parameter $\delta =3$, the e-folds number reaches the values $50$
and $60$ in consistency with the contour plot of the $(r,n_{s})$ plane obtained
by Planck data at $1\sigma $ and $2\sigma $ C.L.. Furthermore, we use the
fraction of the energy density and the variance of the density perturbations
approach to study the abundance of the production PBH. We find that the PBH
overproduction can be satisfied for specific values of parameters of the
non-adiabatic curvature power spectrum at some narrow parametric resonance.
Moreover, to explain the GWs expected by observations, the duration of
preheating should be bounded by values less or equal to $2$. This bound is in
agreement with values of the tensor-to-scalar ratio and the spectral index
constrained by Planck data.
