Vibration suppression has been a widely studied topic for a long time, with various modifications in passive vibration mitigation devices to improve the efficacy. One such modification is the addition of the inerter. The inerter has been integrated into various vibration mitigation devices, whose mass amplification effect could be used to enhance the performance of dynamic vibration absorbers. In the current study, we consider an inerter based pendulum vibration absorber (IPVA) system and conduct a theoretical study on vibration suppression of the device. The IPVA system operates based on the principle of nonlinear energy transfer, wherein the energy of the primary structure is transferred into the pendulum vibration absorber. This is the result of parametric resonance of the pendulum, where the primary resonance of the system becomes unstable and a harmonic regime containing a frequency half the resonant frequency emerges (referred to as secondary regime). We use the harmonic balance method along with bifurcation analysis using Floquet theory to study the stability of primary resonance. It is observed that a pitchfork bifurcation and period-doubling bifurcation are necessary for nonlinear energy transfer to occur. Furthermore, we integrate the IPVA with a linear, harmonically forced oscillator to demonstrate its efficacy compared with a linear benchmark. We also examine the effects of various system parameters on the occurrence of the secondary regime. Moreover, we verify the nonlinear energy transfer phenomenon (due to the occurrence of the secondary regime) by numerical Fourier analysis.