  We study a class of linear parabolic path-dependent PDEs (PPDEs) defined on
the space of c\`adl\`ag paths $x \in D([0,T])$,
  in which the coefficient functions at time $t$ depend on $x(t)$ and
$\int_{0}^{t}x(s)dA_{s}$, for some (deterministic) continuous function $A$ with
bounded variations.
  Under uniform ellipticity and H\"older regularity conditions on the
coefficients, together with some technical conditions on $A$,
  we obtain the existence of a smooth solution to the PPDE by appealing to the
notion of Dupire's derivatives.
  It provides a generalization to the existing literature studying the case
where $A_t = t$,
  and complements our recent work, Bouchard and Tan (2021), on the regularity
of approximate viscosity solutions for parabolic PPDEs.
  As a by-product, we also obtain existence and uniqueness of weak solutions
for a class of path-dependent SDEs.
