Magnetoelastic coupling and uniaxial pressure dependencies of the ferromagnetic ordering temperature in the quasi-two-dimensional layered van der Waals material ${\mathrm{CrI}}_3$ are experimentally studied and quantified by high-resolution dilatometry. Clear anomalies in the thermal expansion coefficients at $T_C$ imply positive (negative) pressure dependencies $\partial T_C/\partial p_i$ for pressure applied along (perpendicular to) the $c$ axis. The experimental results are backed up by numerical studies showing that the dominant, intralayer magnetic coupling increases upon compression along the $c$ direction and decreases with negative in-plane strain. In contrast, interlayer exchange is shown to initially increase and subsequently decrease upon the application of both out-of-plane and in-plane compression.