This paper is devoted to continuity results of the time derivative of the solution to the one-dimensional parabolic obstacle problem with variable coefficients. Under regularity assumptions on the obstacle and on the coefficients, we prove that the time derivative of the solution is continuous for almost every time. When the solution is nondecreasing in time this result holds for every time. We also give an energy criterion which characterizes the continuity of the time derivative of the solution at a point of the free boundary. Such a problem arises in the pricing of american options in generalized Black-Scholes models of finance. Our results apply in financial mathematics.