A study of prompt charm-hadron pair production in proton-lead collisions at sqrt[s_{NN}]=8.16 TeV is performed using data corresponding to an integrated luminosity of about 30 nb^{-1}, collected with the LHCb experiment. Production cross sections for different pairs of charm hadrons are measured and kinematic correlations between the two charm hadrons are investigated. This is the first measurement of associated production of two charm hadrons in proton-lead collisions. The results confirm the predicted enhancement of double parton scattering production in proton-lead collisions compared to the single parton scattering production.