This article establishes alcove walk models for intersections of Schubert varieties and partially semi-infinite orbits in the affine Grassmannian of a split reductive group (we call such intersections parabolic Mirković-Vilonen intersections). More precisely, we describe explicit cellular pavings of these intersections, indexed by certain positively-folded alcove walks. We prove a parametrization of the irreducible components of maximal possible dimension, in terms of alcove walks of maximal possible dimension. We then deduce a new combinatorial description of branching to Levi subgroups of irreducible highest weight representations, and in particular we give a new algorithm for computing the characters of such representations.