  We consider mass-deformed theories with ${\cal N}\geq2$ supersymmetry on
round and squashed three-spheres. By embedding the supersymmetric backgrounds
in extended supergravity we show that at special values of mass deformations
the supersymmetry is enhanced on the squashed spheres. When the $3d$ partition
function can be obtained by a limit of a $4d$ index we also show that for these
special mass deformations only the states annihilated by extra supercharges
contribute to the index. By using an equivalence between partition functions on
squashed spheres and ellipsoids, we explain the recently observed squashing
independence of the partition function of mass-deformed ABJ(M) theory on the
ellipsoid. We provide further examples of such simplification for various $3d$
supersymmetric theories.
