This paper is concerned with the numerical solution of symmetric large-scale Lyapunov equations with low-rank right-hand sides and coefficient matrices depending on a parameter. Specifically, we consider the situation when the parameter dependence is sufficiently smooth, and the aim is to compute solutions for many different parameter samples. On the basis of existing results for Lyapunov equations and parameter-dependent linear systems, we prove that the tensor containing all solution samples typically allows for an excellent low multilinear rank approximation. Stacking all sampled equations into one huge linear system, this fact can be exploited by combining the preconditioned CG method with low-rank truncation. Our approach is flexible enough to allow for a variety of preconditioners based, for example, on the sign function iteration or the alternating direction implicit method.