In this article, we propose a new frailty model based on a mixture of inverse Gaussian distributions for multivariate lifetimes. This approach provides an advantage over previous models, as the weights are directly determined through parameterization of the mixture, removing the need for arbitrary guesswork in the weighting process. Moreover, the closed-form Laplace transform of the model facilitates the quantification of Kendall's tau measure of dependence. The frailty model's parametric and flexible parametric variants are examined. For parameter estimation, the expectation-maximization technique is employed, taking advantage of the hierarchical representation of the frailty distribution, providing a simpler and more stable method than directly maximizing the observed log-likelihood function. The performance of the estimators is assessed numerically using Monte Carlo simulations. We apply our methodology to two medical datasets on cancer. The results indicate the benefits of the proposed model over existing frailty models in the literature. The implementation of the procedure is added to the R package extrafrail.