A new mechanism for the instability of steady mountain waves is found through analysis of the linear stability problem. Steady flow of a hydrostatic stratified fluid is known to be unstable when the streamlines are at, or very close to, overturning. When the topography has multiple peaks, it is shown that this criterion can be superseded by an instability owing to a resonant triad interaction. For flow over two peaks, the threshold heights for instability are roughly half those which produce overturning streamlines. The mechanism behind the instability is the parametric amplification of counter-propagating gravity waves. The resonant nature of the instability is further illustrated by the existence of discrete peak-to-peak separation distances where the growth rate is a maximum.