We study the stability of large amplitude, circularly polarized Alfvén waves in an anisotropic plasma described by the double-adiabatic/CGL closure, and in particular the effect of a background thermal pressure anisotropy on the well-known properties of Alfvén wave parametric decay in Magnetohydrodynamics (MHD). Anisotropy allows instability over a much wider range of values of parallel plasma beta (β ) when ξ = p 0⊥ /p 0 > 1. When the pressure anisotropy exceeds a critical value, ξ ≥ ξ * with ξ * 2.7, there is a new regime in which the parametric instability is no longer quenched at high β and in the limit β 1 the growth rate becomes independent of β . In the opposite case of ξ < ξ * , the instability is strongly suppressed with increasing parallel plasma beta, similarly to the MHD case. We analyze marginal stability conditions for parametric decay in the (ξ, β ) parameter space, and discuss possible implications for Alfvénic turbulence in the solar wind. arXiv:1711.06371v1 [physics.space-ph]