Recently, the Hilbert–Schmidt speed, as a special class of quantum statistical speed, has been reported to improve interferometric phase in a single-parameter quantum estimation. Here, we test this concept in the multiparameter scenario where two laser phases are estimated in a theoretical model consisting of a three-level atom interacting with two classical monochromatic fields. When the atom is initially prepared in the lower bare state, taking into account the detuning parameters, we extract an exact analytical solution of the atomic density matrix in the case of a two-photon resonant transition. Further, we compare the performance of laser phase parameter estimation in individual and simultaneous metrological strategies and explore the role of quantum coherence in improving the efficiency of unknown multiphase shift estimation protocols. The obtained results show that the Hilbert–Schmidt speed detects the lower bound on the statistical estimation error as well as the optimal estimation regions, where its maximal corresponds to the maximal quantum Fisher information; further, the performance of simultaneous multiparameter estimation with individual estimation inevitably depends on the detuning parameters of the three-level atom. Aside from the quantum entanglement, the quantum coherence is also a crucial resource to improve the accuracy of a metrological protocol.