Wideband spectrum sensing is one of the core components of cognitive radio. A novel frugal sensing scheme was recently proposed by Mehanna et al, aiming to crowdsource spectrum sensing operations to a network of sensors transmitting randomly filtered power measurement bits to a fusion center (FC). The ambient power spectrum is then estimated at the FC using a non-parametric approach. Here, it is assumed that the primary signal admits a Moving Average (MA) parametrization, and the frugal sensing problem is revisited from a parametric spectral estimation point of view. We show that the problem of estimating admissible MA parameters (and thus the MA power spectrum) from single bit quantized data can be formulated as a non-convex Quadratically Constrained Quadratic Program (QCQP). This is NP-Hard in general, but semidefinite-relaxation (SDR) can be employed to obtain approximate solutions. Simulations reveal the superior performance of the SDR technique over the globally optimal solution obtained from the non-parametric formulation, when the MA assumption is valid.