We report the results of the parquet renormalization group (RG) analysis of the phase diagram of the most general 5-pocket model for Fe-based superconductors. We use as an input the orbital structure of excitations near the five pockets made out of dxz, dyz, and dxy orbitals and argue that there are 40 different interactions between low-energy fermions in the orbital basis. All interactions flow under the RG, as one progressively integrates out fermions with higher energies. We find that the low-energy behavior is amazingly simple, despite the large number of interactions. Namely, at low energies the full 5-pocket model effectively reduces either to a 3-pocket model made of one dxy hole pocket and two electron pockets or a 4-pocket model made of two dxz/dyz hole pockets and two electron pockets. The leading instability in the effective 4-pocket model is a spontaneous orbital (nematic) order, followed by s+- superconductivity. In the effective 3-pocket model, orbital fluctuations are weaker, and the system develops either s+- superconductivity or a stripe spin-density wave. In the latter case, nematicity is induced by composite spin fluctuations.