  We consider the production of $W$-boson plus dijet, $W$-boson plus b-jets and
same sign $WW$ via double parton scattering (DPS) in $pA$ collisions at the LHC
and evaluate the corresponding cross sections. The impact of a novel DPS
contribution pertinent to $pA$ collisions is quantified. Exploiting the
experimental capability of performing measurements differential in the impact
parameter in $pA$ collisions, we discuss a method to single out this novel DPS
contribution. The method allows the subtraction of the leading twist (LT)
single parton scattering background and it gives access in a very clean way to
double parton distribution functions in the proton. We calculate leading twist
and DPS cross sections and study the dependence of the observables on the cuts
in the jets phase space. In the $Wjj$ channel the observation of DPS is
possible with data already accumulated in $pA$ runs and the situation will
greatly improve for the next high luminosity runs. For the $Wb\bar{b}$ final
state, the statistics within the current data is too low, but there is
possibility to observe DPS in this channel in future runs, albeit with much
reduced sensitivity than in $Wjj$ final state.
  Finally the DPS observation in the same sign $WW$ channel will require either
significant increase of integrated luminosity beyond that foreseen in next $pA$
runs or improved methods for $W$ reconstruction, along with its charge, in
hadronic decay channels.
