A simple analytical model is developed to analyze and explain the complex dynamics of the multipeak modulation instability spectrum observed in dispersion oscillating optical fibers [M. Droques et al., Opt. Lett. 37, 4832 (2012)]. We provide a simple expression for the local parametric gain, which shows that each of the multiple spectral components grows thanks to a quasi-phase-matching mechanism due to the periodicity of the waveguide parameters, in good agreement with numerical simulations and experiments. This simplified model is also successfully used to tailor the multipeak modulation instability spectrum shape. These theoretical predictions are confirmed by experiments.