Observations of exotic structures in the J/ψp channel, which we refer to as charmonium-pentaquark states, in Λ_{b}^{0}→J/ψK^{-}p decays are presented. The data sample corresponds to an integrated luminosity of 3 fb^{-1} acquired with the LHCb detector from 7 and 8 TeV pp collisions. An amplitude analysis of the three-body final state reproduces the two-body mass and angular distributions. To obtain a satisfactory fit of the structures seen in the J/ψp mass spectrum, it is necessary to include two Breit-Wigner amplitudes that each describe a resonant state. The significance of each of these resonances is more than 9 standard deviations. One has a mass of 4380±8±29 MeV and a width of 205±18±86 MeV, while the second is narrower, with a mass of 4449.8±1.7±2.5 MeV and a width of 39±5±19 MeV. The preferred J^{P} assignments are of opposite parity, with one state having spin 3/2 and the other 5/2.