Two rigid plates are vertically suspended by thread such that they are parallel to and opposite each other. The plates are partially submerged in a dish of liquid that is attached to the top of a vertical shake table. When the shake table is driven with noise in a frequency band, random surface waves are parametrically excited, and the plates move toward each other. The reason for this attraction is that the waves carry momentum, and the wave motion between the plates is visibly reduced. The behavior is analogous to the Casimir effect, in which two conducting uncharged parallel plates attract each other due to the zero-point spectrum of electromagnetic radiation. The water wave analog can be readily demonstrated and offers a visual demonstration of a Casimir-type effect. Measurements of the force agree with the water wave theory even at large wave amplitudes, where the theory is expected to break down. The water wave analog applies to side-by-side ships in a rough sea and is distinct from the significant attraction that can be caused by a strong swell.