We examine vibrational resonance (VR) in a position-dependent mass (PDM) oscillator with doubly-singular mass distribution function describing the vibrational inversion mode of NH3 molecule. The impacts of the PDM parameters on VR was studied by computing the response amplitudes as functions of the amplitude of high-frequency component of the dual-frequency driving forces and the PDM parameters. We show for the first time that, beside the significant roles played by the parameters of the variable mass in inducing and controlling resonances similar to the forcing parameters, the variable mass parameters impacts on the resonance characteristics by leading the system from single resonance into double resonance.