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Cerebrospinal flow dynamics (CSF) plays a critical role in structural disorders of the central nervous system (CNS) and in the design of effective procedures for intrathecal drug delivery. Medical imaging techniques have only partially characterized CSF dynamics. Computational models have the potential to offer a high-resolution description of CSF flow and advance our mechanistic understanding. However, anatomically-accurate computational models of CSF dynamics in the spinal canal have largely ignored the compliance of the spinal tissues, which is critical to understand the pulse wave velocity and the craniocaudal decay of CSF pulsations. Here, we propose a mixed-dimensional fluid-structure interaction method that enables high-fidelity simulations of CSF dynamics on anatomically-accurate models of the spinal canal, considering the tissue compliance effects emerging from the dura mater and epidural fat. Our mixed-dimensional approach bypasses a critical computational bottleneck that emerges from the multiscale geometry of spinal tissues. Our results show that accurate modeling of tissue compliance is critical to capture key elements of CSF dynamics. This work opens new possibilities to control and optimize intrathecal drug delivery and to understand structural abnormalities of the CNS.