  We use deep Hubble Space Telescope imaging of the Frontier Fields to
accurately measure the galaxy rest-frame ultraviolet luminosity function (UV
LF) in the redshift range $z \sim 6-8$. We combine observations in three
lensing clusters A2744, MACS0416, MACS0717 and their associated parallels
fields to select high-redshift dropout candidates. We use the latest lensing
models to estimate the flux magnification and the effective survey volume in
combination with completeness simulations performed in the source plane. We
report the detection of 227 galaxy candidates at $z=6-7$ and 25 candidates at
$z \sim 8$. While the total survey area is about 4 arcmin$^{2}$ in each
parallel field, it drops to about 0.6 to 1 arcmin$^{2}$ in the cluster core
fields because of the strong lensing. We compute the UV luminosity function at
$z \sim 7$ using the combined galaxy sample and perform Monte Carlo simulations
to determine the best fit Schechter parameters. We are able to reliably
constrain the LF down to an absolute magnitude of $M_{UV}=-15.25$, which
corresponds to 0.005$L^{\star}$. More importantly, we find that the faint-end
slope remains steep down to this magnitude limit with
$\alpha=-2.04_{-0.17}^{+0.13}$. Our results confirm the most recent results in
deep blank fields but extend the LF measurements more than two magnitudes
deeper. The UV LF at $z \sim 8$ is not very well constrained below $M_{UV}=-18$
due to the small number statistics and incompleteness uncertainties. To assess
the contribution of galaxies to cosmic reionization we derive the UV luminosity
density at $z\sim7$ by integrating the UV LF down to an observationally
constrained limit of $M_{UV} = -15$. We show that our determination of
Log($\rho_{UV}$)=$26.2\pm0.13$ (erg s$^{-1}$ Hz$^{-1}$ Mpc$^{-3}$) can be
sufficient to maintain the IGM ionized.
