Continuous liquid interface production (CLIP), a form of vat photopolymerization (VP), affords designers unprecedented geometric complexity in their products; however, currently CLIP requires cumbersome support structures that are materially wasteful, human labor-intensive, time-consuming to remove, and damaging to part surface finish. Recent physical demonstrations suggest injection through conduits embedded within the part itself can offset suction forces during CLIP. Here we demonstrate that in addition to increasing printing speeds, such injection can alleviate the need for supports. To demonstrate such improvements, we develop a novel fluid dynamics-guided computational inverse design tool, Paraflow. Paraflow takes as input a user's arbitrary 3D model for printing, formulates the design problem as a path planning optimization problem, and computationally designs a corresponding fluidic injection network that distributes one, or potentially multiple, materials during printing. We experimentally show such networks enable printing of farther unsupported overhang geometries than can current state-of-the-art CLIP methods.