Uplink sensing is still a relatively unexplored scenario in integrated sensing and communication which can be used to improve positioning and sensing estimates. We introduce a pilot-based maximum likelihood, and a maximum a posteriori parametric channel estimation procedure using an orthogonal frequency division multiplexing (OFDM) waveform in uplink sensing. The algorithm is capable of estimating the multipath components of the channel, such as the angles of arrival, departure, path coefficient, and the delay and Doppler terms. As an advantage, when compared to other existing methods, the proposed procedure presents expressions for exact alternating coordinate updates, which can be further improved to achieve a competitive multipath channel estimation tool.