From first-principles density functional theory calculations combined with varying temperature Raman experiments, we show that AFe$_2$As$_2$ (A=Ba, Sr), the parent compound of the FeAs based superconductors of the new structural family, undergoes a spin-Peierls-like phase transition at low temperature. The coupling between the phonons and frustrated spins is proved to be the main cause of the structural transition from the tetragonal to orthorhombic phase. These results well explain the magnetic and structural phase transitions in AFe$_2$As$_2$(A=Ba, Sr) recently observed by neutron scattering.
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