  In this article, we extend Sullivan's PL de Rham theory to obtain simple
algebraic models for the rational homotopy theory of parametrised spectra. This
simplifies and complements the results of arXiv:1910.14608, which are based on
Quillen's rational homotopy theory.
  According to Sullivan, the rational homotopy type of a nilpotent space $X$
with finite Betti numbers is completely determined by a commutative
differential graded algebra $A$ modelling the cup product on rational
cohomology. In this article we extend this correspondence between topology and
algebra to parametrised stable homotopy theory: for a space $X$ corresponding
to the cdga $A$, we prove an equivalence between specific rational homotopy
categories for parametrised spectra over $X$ and for differential graded
$A$-modules. While not full, the rational homotopy categories we consider
contain a large class of parametrised spectra. The simplicity of the approach
that we develop enables direct calculations in parametrised stable homotopy
theory using differential graded modules.
  To illustrate the usefulness of our approach, we build a comprehensive
dictionary of algebraic translations of topological constructions; providing
algebraic models for base change functors, fibrewise stabilisations,
parametrised Postnikov sections, fibrewise smash products, and complexes of
fibrewise stable maps.
