This paper is devoted to the existence of random attractors for rough partial differential equations driven by nonlinear multiplicative Hölder rough paths with exponents in (1/3,1/2]. Our approach relies upon rough paths theory and stopping times analysis in a suitable scale of interpolation spaces. The core step is to derive the adequate algebraic and analytical properties of a sequence of stopping times, which allows us to establish the required compact tempered absorbing set. The existence of a pullback attractor for the generated random dynamical system is straightforward. An illustrative example is presented by reaction-diffusion equations subjected to fractional Brownian rough paths.