In this paper we address optimal control problems subject to fluid flows through deformable, porous media. In particular, we focus on linear quadratic elliptic-parabolic control problems, with both distributed and boundary controls, and prove existence and uniqueness of optimal control. Furthermore, we derive the first order necessary optimality conditions. These problems have important biological and biomechanical applications. For example, optimizing the pressure of the flow, and investigating the influence and control of pertinent biological parameters are relevant in the case of the lamina cribrosa – a porous tissue at the base of the optic nerve head inside the eye – which is modeled by poroelasticity – where these factors are believed to be related to the development of ocular neurodegenerative diseases such as glaucoma. Moreover, the study and results will be applicable to other situations such as the poroelastic modeling of cartilages, bones, and engineered tissues.