A simple mechanism of parametric excitation of internal gravity waves in a uniformly stably stratified flow under the inviscid Boussinesq approximation is presented. It consists in an oscillating planar irrotational strain field with frequency $\omega$ disturbed by three-dimensional plane waves. When the amplitude of the strain is weak, the problem is reduced to a Mathieu equation and a condition for parametric resonance is easily deduced. For a large-amplitude strain field equations are solved numerically with Floquet theory. In both cases, it is shown that parametric instabilities are excited when stratification is large enough, that is when $N > \frac{1}{2}\omega$, where $N$ is the Brunt–Väisälä frequency of the flow. On the other hand, when $N \leq \frac{1}{2}\omega$, the flow is shown to be stable for any periodic background excitation thanks to a theorem by Joukowski. Therefore, stratification promotes instability. In the strongly stratified case $N\,{\gg}\,\omega$, resonant waves satisfy the Billant–Chomaz self-similarity law and the resulting instabilities develop inside correlated quasi-horizontal layers. After discussion of the viscous effects, the theory of the paper is applied to the stability of an elliptical vortex in a rotating stratified medium.