  We provide a theoretical description of light scattering by a spherical
particle whose permittivity is modulated in time at twice the frequency of the
incident light. Such a particle acts as a finite-sized photonic time crystal
and, despite its sub-wavelength spatial extent, can host optical parametric
amplification. Conditions of parametric Mie resonances in the sphere are
derived. We show that time-modulated materials provide a route to tailor
directional light amplification, qualitatively different from that in
scatterers made from a gain media. We design two characteristic time-modulated
spheres that simultaneously exhibit light amplification and desired radiation
patterns, including those with zero backward and/or vanishing forward
scattering. The latter sphere provides an opportunity for creating shadow-free
detectors of incident light.
