We study a class of linear parabolic path-dependent PDEs (PPDEs) defined on the space of càdlàg 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 Hölder 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.