We prove well-posedness in $H^σ(\mathbb{R})$ for any $σ\in [0,\infty)$ of a parametrically forced nonlinear Schrödinger equation (PFNLS) in one dimension driven by multiplicative Stratonovich noise which has spatially homogeneous statistics. The noise is white in time and correlated in space. We first construct local mild solutions via a fixed-point argument. We then formulate a blow-up criterion by showing that the equation has persistence of integrability and regularity as long as the $L^2(\mathbb{R})$-norm of the solution remains finite. Afterwards we derive a pathwise estimate on the $L^2(\mathbb{R})$-norm using a mild Itô formula. Our results also apply to the standard cubic NLS equation driven by multiplicative translation-invariant Stratonovich noise.