We estimate the 3 \pi continuum contribution to the nucleon strange quark vector current form factors, including the effect of a 3 \pi <----> \rho \pi resonance. We find the magnitude of this OZI-violating contribution to be comparable to that of typical OZI-allowed contributions. We also study the isoscalar electromagnetic form factors, and find that the presence of a \rho \pi resonance in the multi-pion continuum may generate an appreciable contribution.