The authors study the stability behavior of a non-linear oscillator parametrically excited by a stationary Markov process. They modify the notion of stability by considering \(H(E_ 0)=\sup_{t>0}E(t,E_ 0)\), where \(E(t)=U(x(t))+\dot x(t)^ 2\) is the sum of potential and kinetic energy, rather than the usual \(\sup_{t>0}| \vec x(t,\vec x_ 0)|\), where \(| \vec x(t)|^ 2=x(t)^ 2+\dot x(t)^ 2\). In case of a linear oscillator (with restoring force x) the latter equals E(t). A state (with energy \(E_ 0)\) is stochastically stable, if the stochastic limit of \(H(E_ 0)\) tends to 0 as the energy \(E_ 0=E(0,E_ 0)\) tends to 0. For small damping coefficient \(\beta\) the transition probability is expanded in terms of \(\beta\) and for its leading part a Fokker-Planck equation in the energy variable E is derived. Feller-type stability criteria are given depending on the nature (in Feller's classification) of the energy boundaries \(E=0\) and \(E=\infty\), and (non-exponential) growth rates for the time \(\tau\) (E) of exceeding the energy level E are estimated.