The Nπ^{0}π^{0} decays of positive-parity N^{*} and Δ^{*} resonances at about 2 GeV are studied at ELSA by photoproduction of two neutral pions off protons. The data reveal clear evidence for several intermediate resonances: Δ(1232), N(1520)3/2^{-}, and N(1680)5/2^{+}, with spin parities J^{P}=3/2^{+}, 3/2^{-}, and 5/2^{+}. The partial wave analysis (within the Bonn-Gatchina approach) identifies N(1440)1/2^{+} and the N(ππ)_{S wave} (abbreviated as Nσ here) as further isobars and assigns the final states to the formation of nucleon and Δ resonances and to nonresonant contributions. We observe the known Δ(1232)π decays of Δ(1910)1/2^{+}, Δ(1920)3/2^{+}, Δ(1905)5/2^{+}, Δ(1950)7/2^{+}, and of the corresponding spin-parity series in the nucleon sector, N(1880)1/2^{+}, N(1900)3/2^{+}, N(2000)5/2^{+}, and N(1990)7/2^{+}. For the nucleon resonances, these decay modes are reported here for the first time. Further new decay modes proceed via N(1440)1/2^{+}π, N(1520)3/2^{-}π, N(1680)5/2^{+}π, and Nσ. The latter decay modes are observed in the decay of N^{*} resonances and at most weakly in Δ^{*} decays. It is argued that these decay modes provide evidence for a 3-quark nature of N^{*} resonances rather than a quark-diquark structure.