The stable Dold-Kan correspondence says that for every commutative ring $R$, there is an equivalence of $\infty$-categories between the category $Ch(R)$ of (unbounded) chain complexes of $R$-modules and the category $Mod(HR)$ of module spectra over the Eilenberg-Maclane ring spectrum $HR$. This can be realized as a Quillen equivalence of suitable model categories (see here).
My question is whether there is a parameterized version of this equivalence. Namely, given a based space $B$, there is a way to construct the category of parametrized spectra over $B$ with a suitable model structure (see here). In this setting there is a monoidal structure given by smash product "over $B$". We therefore can construct the parametrized ring spectrum $HR\times B$ and I guess also the category of parametrized modules over it (with suitable model structure). I wonder if this is equivalent (as an $\infty$-category) to something we can construct on the level of chains.
My guess is that it should be equivalent to something like co-modules over the co-algebra $C_* (B)$ in $Ch(R)$. Is there indeed some algebraic model like this?