High-resolution and anatomically realistic computer models of biological soft tissues play a significant role in the understanding of the function of cardiovascular components in health and disease. However, the computational effort to handle fine grids to resolve the geometries as well as sophisticated tissue models is very challenging. One possibility to derive a strongly scalable parallel solution algorithm is to consider finite element tearing and interconnecting (FETI) methods. In this study, we propose and investigate the application of FETI methods to simulate the elastic behavior of biological soft tissues. As one particular example, we choose the artery which isas most other biological tissuescharacterized by anisotropic and nonlinear material properties. We compare two specific approaches of FETI methods, classical and all-floating, and investigate the numerical behavior of different preconditioning techniques. In comparison with classical FETI, the all-floating approach not only has advantages concerning the implementation but also has advantages concerning the convergence of the global iterative solution method. This behavior is illustrated with numerical examples. We present results of linear elastic simulations to show convergence rates, as expected from the theory, and results from the more sophisticated nonlinear case where we apply a well-known anisotropic model to the realistic geometry of an artery. Although the FETI methods have a great applicability on artery simulations, we will also discuss some limitations concerning the dependence on material parameters.