In this talk we will explain how we can associate in a natural way Bernstein processes to a class of non-autonomous linear parabolic initial- and final-boundary value problems defined in suitable open bounded regions of Euclidean space of arbitrary dimension. Under certain conditions regarding their joint endpoint distributions, we will also indicate how those processes become reversible Markov diffusions which in addition satisfy two Itô equations for some suitably constructed Wiener processes. We shall finally focus on two highlights of our construction, namely, on the one hand a relation that expresses the probability density of the Bernstein diffusions in terms of the solutions to the given parabolic equations and, on the other hand, Feynman-Kac representations of those solutions in terms of the Bernstein diffusions.