To obtain subsurface information from onshore seismic exploration data using full-waveform inversion (FWI) based on the acoustic wave equation, elastic waves, such as ground roll and mode-converted waves, should be suppressed through heavy preprocessing. However, such preprocessing deforms not only the elastic waves but also the acoustic waves. Moreover, it is not easy to separate body waves from surface waves in seismic traces. For these reasons, we need to generate both types of waves in the modeling step to obtain seismic waves that are similar to real seismic waves. Therefore, elastic FWI using the elastic wave equation is necessary to achieve a more accurate FWI. In addition, elastic FWI can provide better geologic information than acoustic FWI because it inverts the P-wave velocity, S-wave velocity, and density. Laplace-Fourier-domain elastic FWI is an effective method because it inverts these multiple parameters and can be applied to real seismic data that lack low-frequency components. However, the conventional Laplace-Fourier-domain FWI requires a matrix solver with a huge memory cost to perform the modeling in the Laplace-Fourier-domain, and memory usage becomes more intensive in the elastic case. In the present study, we combined time-domain wave propagation modeling and Laplace-Fourier-domain elastic FWI to invert multiple parameters with less memory cost. By using time-domain modeling, which does not require a matrix solver, we obtained the forward and adjoint wavefields with less memory cost. The residuals between the recorded and modeled data, the virtual sources, the Hessian matrices, and the gradient directions were calculated in the Laplace-Fourier domain. To validate the proposed algorithm, we performed numerical tests with Model 94 synthetic data and Benjamin Creek real seismic data.