Magnon damping is a key factor governing spin-wave transport and nonlinear dynamics of multimode magnon systems. However, many descriptions rely on the assumption of an effective mode-independent parameter, which can mask wavelength-dependent relaxation processes that depend on propagation geometry and mode profile. Here, we employ high-resolution parametric-instability spectroscopy to probe thickness-quantized spin waves with wavelengths down to about a hundred nanometers in micrometer-thick yttrium iron garnet films. The instability threshold exhibits a regular sawtooth dependence on the magnetic field, arising from switching between discrete thickness modes. Comparison with dipole–exchange theory reveals anisotropic wavevector-dependent damping that increases with mode number and depends differently on the in-plane and out-of-plane wavevector components.