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We present a laboratory study on the instability of internal wave attractors in a trapezoidal fluid domain filled with uniformly stratified fluid. Energy is injected into the system via standing-wave-type motion of a vertical wall. Attractors are found to be destroyed by parametric subharmonic instability via a triadic resonance which is shown to provide a very efficient energy pathway from long to short length scales. This Letter provides an explanation of why attractors may be difficult or impossible to observe in natural systems subject to large amplitude forcing.