This work introduces an extension of the high order, single stage Lax-Wendroff Flux Reconstruction (LWFR) scheme to solve second order time-dependent partial differential equations in a conservative form on curvilinear meshes. The method uses the BR1 scheme to reduce the system to first order so that the LWFR scheme can be applied. The work makes use of embedded error-based time stepping recently introduced for LWFR schemes which becomes particularly relevant in the absence of a CFL stability limit for parabolic equations. The scheme is verified to show optimal order of convergence and validated with transonic flow over airfoil and unsteady flow over cylinder.