The parameterization of anisotropic models is very important when focusing on specific signatures of seismic waves and reducing the parameter crosstalk involved in inverting seismic data. The parameterization is strongly dependent on the problem at hand. We have developed a new parameterization for an elastic orthorhombic (ORT) model with on-axes compressional wave (P-wave) and shear wave (S-wave) velocities and new symmetric anelliptic parameters. The perturbation approach is well defined for P-waves in acoustic ORT media. In the elastic ORT media, the P-wave perturbation coefficients are very similar to their acoustic counterparts. However, the S-wave perturbation coefficients are still unknown. The perturbation coefficients can be interpreted as sensitivity coefficients, and they are important in many applications. We apply the second-order perturbation in anelliptic parameters for P-, S1-, and S2-wave phase velocities in the elastic ORT model. We find that when using the conventional method some perturbation coefficients for S-waves are not defined in the vicinity of the singularity point in an elliptical background model. Thus, we adopt an alternative perturbation approach that overcomes this problem. We compute the first- and second-order perturbation coefficients for P- and S-waves. The perturbation-based approximations are very accurate for P- and S-waves compared with the exact solutions, based on a numerical example. The reductions to transversely isotropic and acoustic ORT models are also considered for analysis. We also determine how perturbations in anelliptic parameters affect S-wave triplications in an elastic ORT model.