Young protoplanetary discs and the outer radii of active galactic nuclei may be subject to gravitational instability and, as a consequence, fall into a ‘gravitoturbulent’ state. While in this state, appreciable angular momentum can be transported; alternatively, the gas may collapse into bound clumps, the progenitors of planets or stars. In this paper, we numerically characterize the properties of 3D gravitoturbulence, focusing especially on its dependence on numerical parameters (resolution, domain size) and its excitation of small-scale dynamics. Via a survey of vertically stratified shearing-box simulations with PLUTO and RODEO, we find (a) evidence that certain gravitoturbulent properties are independent of horizontal box size only when the box is larger than ≃40H, where H is the scaleheight, (b) at high resolution, small-scale isotropic turbulence appears off the mid-plane around z ≃ 0.5–1H and (c) this small-scale dynamics results from a parametric instability, involving the coupling of inertial waves with a large-scale axisymmetric epicyclic mode. This mode oscillates at a frequency close to Ω and is naturally excited by gravitoturbulence via a non-linear process to be determined. The small-scale turbulence we uncover has potential implications for a wide range of disc physics, e.g. turbulent saturation levels, fragmentation, turbulent mixing and dust settling.