A Luenberger type dynamical observer is proposed for estimation of the full state of a nonautonomous linear parabolic equation, from a finite-dimensional output. The output is given by average-like measurements of the state on small regions. The state estimate given by the observer converges exponentially to the real state, as time increases. The operator which injects the output into the dynamics of the observer is explicit and involves an appropriate oblique projection. Results of numerical simulations are presented showing the exponential stability of the error dynamics, and thus showing the estimating performance of the observer.