The prompt D^{0} meson azimuthal anisotropy coefficients, v_{2} and v_{3}, are measured at midrapidity (|y|<1.0) in Pb-Pb collisions at a center-of-mass energy sqrt[s_{NN}]=5.02 TeV per nucleon pair with data collected by the CMS experiment. The measurement is performed in the transverse momentum (p_{T}) range of 1 to 40 GeV/c, for central and midcentral collisions. The v_{2} coefficient is found to be positive throughout the p_{T} range studied. The first measurement of the prompt D^{0} meson v_{3} coefficient is performed, and values up to 0.07 are observed for p_{T} around 4 GeV/c. Compared to measurements of charged particles, a similar p_{T} dependence, but smaller magnitude for p_{T}<6 GeV/c, is found for prompt D^{0} meson v_{2} and v_{3} coefficients. The results are consistent with the presence of collective motion of charm quarks at low p_{T} and a path length dependence of charm quark energy loss at high p_{T}, thereby providing new constraints on the theoretical description of the interactions between charm quarks and the quark-gluon plasma.