Elastic anisotropy has a critical impact on the mechanical behavior of rocks, which plays an important role in unconventional and fractured reservoirs for seismic interpretation, well completion design, and determination of hydraulic fracturing parameters. Most unconventional rocks are described as vertical transversely isotropic (VTI) materials. A common practice used to estimate the elastic properties of VTI rocks and their dependence on in-situ stress is using triaxial frames. This practice requires the acquisition of three ultrasonic elastic wave measurements using three independent cores obtained from the same interval in a well and with bedding angles of 0°, 45° and 90° with respect to the axis of symmetry of the VTI samples. The corresponding five independent Thomsen's parameters (V_P0, V_S0, Epsilon, Delta and Gamma) are estimated from the compressional and shear arrivals at these three angles while the sample undergoes confining and deviatoric stress. Because the number of cores in a well is limited, we develop a method to estimate the stress-dependent V_P0, Epsilon and Delta, using a single core placed inside a triaxial frame. First, we measure phase and group velocities using ultrasonic transducers while the rock undergoes confining and deviatoric stress. Phase velocities are obtained using conventional ultrasonic end-caps in the triaxial frame while group velocities are obtained by adding small-size transducers around the core sample. The small-size transducers work both as receivers and transmitters, generating a waveform dataset with tomographic coverage. Then, we apply a Bayesian inversion method to calculate V_P0, Epsilon and Delta using the group velocity measurements. We combine inversion results with phase measurements to estimate the stress-dependent velocity anisotropy. The developed method accurately captures anisotropy in a single horizontal core under conditions of stress. We test the method by applying isotropic and deviatoric stress paths on two samples: a Berea sandstone with negligible initial anisotropy and a Mancos shale with a high degree anisotropy. Results show that the elastic response from velocity and strains is linked to pre-existing anisotropy within the rock. Inverted anisotropy parameters are consistent with previous work where anisotropy parameters were obtained using the conventional method that requires three cores.