In this study, the joint source-and-relay power allocation (JPA) problem in multiple-input multiple-output (MIMO) relay systems is investigated. It is revealed for the first time that the JPA problem based on the mean squared error (MSE) minimisation or the achievable rate maximisation criterion possesses a partial convexity, that is, the cost function is convex with respect to (w.r.t.) part of the optimisation parameters only. It is then proved that a better partial convexity, namely, a partial convexity w.r.t. a larger subset of the optimisation parameters, can be achieved by relaxing the cost function with an upper or lower bound of the MSE or achievable rate. By exploiting the partial convexity properties disclosed, two iterative algorithms are proposed to solve the non-convex JPA problem. It is shown that the convergence of the proposed iterative algorithms is guaranteed because of the fact that each iteration follows a convex optimisation w.r.t. the partial parameters. Finally, it is shown by Monte Carlo simulations that the proposed algorithms outperform several existing methods including our own prior work.