We measure the dispersion of the Cu-O bond-stretching phonon mode in the high-temperature superconducting parent compound ${\mathrm{Ca}}_2{\mathrm{CuO}}_2{\mathrm{Cl}}_2$. Our density functional theory calculations predict a cosine-shaped bending of the dispersion along both the $(\xi 00$) and $(\xi \xi 0$) directions, while comparison with previous results on ${\mathrm{Ca}}_{1.84}{\mathrm{CuO}}_2{\mathrm{Cl}}_2$ show it only along $(\xi 00$), suggesting an anisotropic effect which is not reproduced in calculation. Comparison with isostructural ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ and literature suggests that these calculations reproduce well the dispersion in the overdoped regime.