  In this paper, we investigate a semilinear stochastic parabolic equation with
a linear rough term $du_{t}=\left[L_{t}u_{t}+f\left(t,
u_{t}\right)\right]dt+\left(G_{t}u_{t}+g_{t}\right)d\mathbf{X}_{t}+h\left(t,
u_{t}\right)dW_{t}$, where $\left(L_{t}\right)_{t \in \left[0, T\right]}$ is a
family of unbounded operators acting on a monotone family of interpolation
Hilbert spaces, $\mathbf{X}$ is a two-step $\alpha$-H\"older rough path with
$\alpha \in \left(1/3, 1/2\right]$ and $W$ is a Brownian motion. Existence and
uniqueness of the mild solution are given through the stochastic controlled
rough path approach and fixed-point argument. As a technical tool to define
rough stochastic convolutions, we also develop a general mild stochastic sewing
lemma, which is applicable for processes according to a monotone family.
