Wideband spectrum sensing is a fundamental component of cognitive radio and other applications. A novel frugal sensing scheme was recently proposed as a means of crowdsourcing the task of spectrum sensing. Using a network of scattered low-end sensors transmitting randomly filtered power measurement bits to a fusion center, a non-parametric approach to spectral estimation was adopted to estimate the ambient power spectrum. Here, a parametric spectral estimation approach is considered within the context of frugal sensing. Assuming a Moving-Average (MA) representation for the signal of interest, the problem of estimating admissible MA parameters, and thus the MA power spectrum, from single bit quantized data is formulated. This turns out being a non-convex quadratically constrained quadratic program (QCQP), which is NP--Hard in general. Approximate solutions can be obtained via semi-definite relaxation (SDR) followed by randomization; but this rarely produces a feasible solution for this particular kind of QCQP. A new Sequential Parametric Convex Approximation (SPCA) method is proposed for this purpose, which can be initialized from an infeasible starting point, and yet still produce a feasible point for the QCQP, when one exists, with high probability. Simulations not only reveal the superior performance of the parametric techniques over the globally optimum solutions obtained from the non-parametric formulation, but also the better performance of the SPCA algorithm over the SDR technique.