This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic partial differential equation (PDE). We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on sigma(j) with j is an element of N, and that the PDE operator may depend nonaffinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [A. Cohen, R. DeVore, and Ch. Schwab, Found. Comput. Math., 10 (2010), pp. 615-646; Anal. Appl., 9 (2011), pp. 1-37], we show that the state and the control are analytic as functions depending on these parameters sigma(j). We establish sparsity of generalized polynomial chaos (gpc) expansions of both state and control in terms of the stochastic coordinate sequence sigma = (sigma(j)) j >= 1 of the random inputs, and we prove convergence rates of best N-term truncations of these expansions. Such truncations are the key for subsequent computations since they do not assume that the stochastic input data has a finite expansion. In a follow-up paper [A. Kunoth and Ch. Schwab, Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained control problems, in preparation], we explain two methods for how such best N-term truncations can be computed practically: by greedy-type algorithms as in [Ch. Schwab and C. J. Gittelson, Acta Numer., 20 (2011), pp. 291-467; C. J. Gittelson, Report 2011-12, Seminar for Applied Mathematics, ETH Zurich, Zurich, Switzerland, 2011], or by multilevel Monte-Carlo methods as in [F. Y. Kuo, Ch. Schwab, and I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351-3374]. The sparsity result allows, in conjunction with adaptive wavelet Galerkin schemes, for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [W. Dahmen and A. Kunoth, SIAM J. Control Optim., 43 (2005), pp. 1640-1675; M. D. Gunzburger and A. Kunoth, SIAM J. Control. Optim., 49 (2011), pp. 1150-1170; A. Kunoth, Numer. Algorithms, 39 (2005), pp. 199-220]; see [A. Kunoth and Ch. Schwab, Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained control problems, in preparation].