Fibrillar hydrogels are remarkably stiff, low-density networks that can hold vast amounts of water. These hydrogels can easily be made anisotropic by orienting the fibrils using different methods. Unlike the detailed and established descriptions of polymer gels, there is no coherent theoretical framework describing the elastoplastic behavior of fibrillar gels, especially concerning anisotropy. In this work, the swelling pressures of anisotropic fibrillar hydrogels made from cellulose nanofibrils were measured in the direction perpendicular to the fibril alignment. This experimental data was used to develop a model comprising three mechanical elements representing the network and the osmotic pressure due to non-ionic and ionic surface groups on the fibrils. At low solidity, the stiffness of the hydrogels was dominated by the ionic swelling pressure governed by the osmotic ingress of water. Fibrils with different functionality show the influence of aspect ratio, chemical functionality, and the remaining amount of hemicelluloses. This general model describes physically crosslinked hydrogels comprising fibrils with high flexural rigidity - that is, with a persistence length larger than the mesh size. The experimental technique is a framework to study and understand the importance of fibrillar networks for the evolution of multicellular organisms, like plants, and the influence of different components in plant cell walls.