The principle of quantitative acoustic microscopy (QAM) is to form two-dimensional (2D) acoustic parameter maps from a collection of radiofrequency (RF) signals acquired by raster scanning a biological sample. Despite their relatively simple structure consisting of two main reflections, RF signals are currently sampled at very high frequencies, e.g., at 2.5 GHz for QAM system employing a single-element transducer with a center frequency of 250-MHz. The use of such high sampling frequencies is challenging because of the potentially large amount of acquired data and the cost of the necessary analog to digital converters. Based on a parametric model characterizing QAM RF signals, the objective of this paper is to use the finite rate of innovation (FRI) framework in order to significantly reduce the number of acquired samples. These are then used to compute Fourier coefficients that are directly fed into a state-of-theart autoregressive (AR)-based method to estimate the model parameters, which finally leads to the reconstruction of accurate 2D maps. The combination of FRI and AR model for sampling and parametric map recovery allows decreasing the required number of samples per RF signal up to a factor of 18 compared to a conventional approach, with a minimal accuracy loss of quantitative acoustic maps, as proven by visual evaluations and numerical results, i.e. PSNR of 24:50 dB and 24:51 dB for the reconstructed speed of sound map and acoustic impedance map respectively.