Random projections (RPs) are known to provide promising results in the context of high-dimensional supervised classification, but even when information on the class membership is not available, the classification issue can be addressed by exploiting the general idea of RP ensemble. In this work, we address the problem of clustering high-dimensional data in the Gaussian mixture model (GMM) framework by resorting to a RP-based ensemble of low-rank estimates for the group-specific covariance matrix. When the number of features is large compared to the number of units, the most widely used solution for GMM estimation employs parsimonious covariance parameterizations via spectral decomposition, in order to cope with possibly singular group-specific covariance matrices. Our approach offers an alternative to these parsimonious parameterizations which guarantees a full rank estimate for each covariance matrix, thus allowing for the estimation of mixtures of Gaussian graphical models, too. The potential of the proposal is demonstrated through a real data application.