We present a complete set of elastic and structural parameters for wurtzite and zinc-blende group-III nitrides, GaN, InN and AlN, calculated ab initio. A high degree of accuracy is ensured using the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional approach. Both lattice and elastic constants are obtained by means of the stresses acting on the crystal, rather than by a minimization of the lattice energy. Our results show very good agreement with available experimental data. Furthermore, all the independent internal strain parameters of the wurtzite nitrides are reported.