In this paper, we consider a delay-coupled neural oscillator system with inhibitory-to- inhibitory connection. The effects of coupled weights and multiple delays are analyzed. It follows that the increasing coupled weights bring out the multiple coexistences of stable and unstable equilibria by a cascade of pitchfork bifurcations. Moreover, multiple delays control the unstable equilibria to stable periodic solutions. Thus, neural oscillator system exhibits the multiperiodicity coexistences. To this end, we firstly study the equilibria properties and give some equilibria patterns. Analyzing the multiple pitchfork bifurcations of trivial and nontrivial equilibria respectively, we find that there are one, three, five and nine equilibria for the increasing coupled weight. The neural system has a single trivial equilibrium and pairs of synchronous or anti-synchronous nontrivial equilibria, which exhibits the multiple equilibria coexistences. Secondly, employing the corresponding characteristic equation, the effects of coupled delays on stable and unstable equilibria are studied. We find that the stable equilibria keep their stability for the increasing delay, which is called the delay-independent stability. However, for the unstable equilibria, coupled delay suppresses system trajectories near these equilibria into some stable periodic solutions. Therefore, the neural oscillator system exhibits the stability coexistences with multiple equilibria and multiperiodicities. [ABSTRACT FROM AUTHOR]