System of differential equations describing the initial stage of the capture of nonlinear oscillations into the parametric autoresonance is considered. Of special interest are resonant solutions whose amplitude increases without bound with time. The stability of such solutions ensures the physical observability of the autoresonant capture. We study the stability of such solutions with respect to persistent perturbations of white noise type, and we show that under certain restrictions on the intensity of the noise, the capture into parametric autoresonance is preserved with probability tending to one.