Generalized polynomial chaos (gPC), often combined with mono-implicit Runge-Kutta (MIRK) methods, is widely used to solve random ODEs. We investigate the impact of stiffness of random ODEs on the gPC performance. We start by extending pragmatic definitions of stiffness used in deterministic ODEs to their random counterparts. Then we introduce gPC with parallel MIRK schemes to solve random stiff ODEs, in which a suitable parallelism partially alleviates the curse of dimensionality. Our stiffness analysis comprises two parts: (i) the relationship between Jacobians of random ODEs and the corresponding gPC equations and (ii) stiffness of the gPC equations. It provides a direct way to determine whether a random ODE and/or the corresponding gPC equations are stiff. This theoretical analysis plays a key role in designing numerical implementations not only of gPC but also of other methods of stochastic computation, e.g., Monte Carlo simulations and stochastic collocation. A series of computational experiments is used to demonstrate the agreement between our theoretical analysis and numerical results and to establish gPC with parallel MIRK as a feasible and effective tool for solving stiff random ODEs.