We study the convergence of semilinear parabolic stochastic evolution equations, posed on a sequence of Banach spaces approximating a limiting space and driven by additive white noise projected onto the former spaces. Under appropriate uniformity and convergence conditions on the linear operators, nonlinear drifts and initial data, we establish convergence of the associated mild solution processes when lifted to a common state space. Our framework is applied to the case where the limiting problem is a stochastic partial differential equation whose linear part is a generalized Whittle-Matérn operator on a manifold $\mathcal{M}$, discretized by a sequence of graphs constructed from a (random) point cloud. In this setting we obtain various discrete-to-continuum convergence results for solutions lifted to $L^q(\mathcal{M})$ for $q \in [2,\infty]$, one of which recovers the $L^\infty$-convergence of a finite-difference discretization of certain (fractional) stochastic Allen-Cahn equations.