We consider a system of two nonlinear differential equations describing the capture into autoresonance in nonlinear oscillators under small parametric driving. Solutions with an infinitely growing amplitude are associated with the autoresonance phenomenon. Stability of such solutions is of great importance because only stable solutions correspond to physically observable motions. We study stability of autoresonant solutions with power asymptotics and show that the random fluctuations of the driving cannot destroy the capture into the parametric autoresonance.