Shock tracking aims to generate a mesh such that element faces align with shock surfaces and other non-smooth features to perfectly represent them with the inter- element jumps in the solution basis, e.g., in the context of a finite volume or discontinuous Galerkin (DG) discretization [1,2]. These methods lead to high-order approximations of high-speed flows and do not require nonlinear stabilization or extensive refinement in non-smooth regions because, once the non-smooth features are tracked by the mesh, the high-order solution basis approximates the remaining smooth features. Implicit shock tracking methods recast the geometrically complex problem of generating a mesh that conforms to all discontinuity surfaces as a PDE-constrained optimization problem [1,2]. The optimization problem seeks to determine the flow solution and nodal coordinates of the mesh that simultaneously minimize an error-based indicator function and satisfy the discrete flow equations. For problems with shocks attached to surfaces, implicit shock tracking requires nodes to slide along surfaces of the object, which can be challenging for complex geometries. In this work, we develop a parametric description of surfaces of the domain from a high-order surface mesh and directly optimize the parametric coordinates of each surface node in the shock tracking setting. This ensures all surface nodes will conform to their original boundaries throughout the optimization procedure, and successfully produces shock-aligned meshes for flows with attached shocks.