A linear backward stochastic partial differential equation \[ u(t,x)= g(x)+ \int ^{T}_{t}\bigl \{(Lu)(s,x)+ (Mq)(s,x)+ f(s,x)\bigr \} ds- \int ^{T}_{t} \langle q(s,x), dW_{s}\rangle , \tag{1} \] \(0\leq t\leq T\), \(x\in \mathbb R^{n}\), is considered, where \(W\) denotes a standard \(d\)-dimensional Wiener process and \[ \begin{aligned} (Lu)(t,x,\omega) &= \nabla (A(t,\omega)\nabla u(t,x,\omega))+ \langle a(t,x,\omega),\nabla u(t,x,\omega)\rangle+ a_{0}(t,x,\omega)u(t,x,\omega),\\ (Mq)(t,x,\omega)& = \text{Tr}\{B(t,\omega)^{T}\nabla q(t,x,\omega)\}+ \langle b(t,x,\omega),q(t,x,\omega)\rangle \end{aligned} \] are second- and first-order differential operators with random coefficients, respectively. Existence, uniqueness and smoothness in \(x\) of a solution \((u,q)\) to (1) is established under some assumptions on the regularity of the (random) functions \(f\), \(g\) and the coefficients of the operators \(L\), \(M\), and the parabolicity assumption \[ A(t)- \tfrac 12 B(t)B(t)^{T}\geq 0 \text{ almost surely for any }t\in [0,T]. \tag{2} \] Let us note that in previous papers on related topics uniform positivity of the matrix \(A(t) -\frac 12 B(t)B(t)^{T}\) was usually supposed instead of (2), cf. e.g. G. Tessitore [Stochastic Anal. Appl. 14, No. 4, 461-486 (1996; Zbl 0876.60044)]. These results are applied to certain forward-backward stochastic differential equations with random coefficients, whose solutions are then related to solutions of (1) via a stochastic Feynman-Kac formula. Finally, applications to option pricing are discussed.