We show that strong parametric driving of a quantum harmonic oscillator coupled to a thermal bath allows one to distinguish between different microscopic models for the oscillator-bath coupling. We consider a bath with an Ohmic spectral density and a model where the system-bath interaction can be tuned continuously between position and momentum coupling via the coupling angle α. We derive a master equation for the reduced density operator of the oscillator in Born-Markov approximation and investigate its quasisteady state as a function of the driving parameters, the temperature of the bath and the coupling angle α. We find that the driving introduces a strong dependence of the time-averaged variance of position and momentum on these parameters. In particular, we identify parameter regimes that maximize the α dependence and provide an intuitive explanation of our results.