Motivated by the need for timely communication and real-time applications, this work considers the problem of characterizing the peak age of information (PAoI) of transmitting through a time-varying first-order Markov fading channel. The communication setting under investigation includes a transmitter, which sends status updates in the form of a coded packet using a linear block code. The receiver estimates the channel using pilot symbols, which are separated by data symbols. To that end, we first derive a semi-analytic expression for the average probability of the correct packet reception over a stationary transmission block of packets. Based on this, we deduce the average PAoI under zero-wait policy. The analytic results show that the PAoI depends on the packet length and the pilot separation in an intriguing way. Specifically, both parameters exhibit two competing effects that influence the PAoI. This suggests the existence of optimal values for packet length and the pilot separation. Moreover, the results show that, in general, neither maximizing the data rate nor minimizing the packet error rate would necessarily minimize the PAoI.