Based on the HSS constitutive model, a three-dimensional finite element model of a tripod bucket foundation in a soft clay seabed was established; the bearing characteristics of the tripod bucket foundation under horizontal cyclic loading were analyzed, and a multi-parameter sensitivity analysis on the accumulated deformation of the foundation was conducted using the range analysis method. The results indicate that during cyclic loading, the axial force transmitted to the top of the buckets via the tripod frame contributes the most to resisting the external total overturning moment. Under the serviceability limit state, the foundation stiffness degrades to 96.3% of its initial value, and the vertical displacement difference between the front and rear buckets is relatively small, resulting in a slow growth rate of the accumulated rotation angle. Under the fatigue limit state, the foundation stiffness degrades to 86.5% of its initial value; the vertical displacement difference between the front and rear buckets increases correspondingly, and the accumulated rotation angle continuously increases with the number of cycles. As the spacing between the foundation buckets increases, the rotation center becomes deeper, and the foundation tends to exhibit a translational movement. The accumulated deformation of the foundation is highly sensitive to the loading frequency and bucket spacing, followed by the initial shear modulus of the soil and the aspect ratio of the buckets, while it is least sensitive to the threshold shear strain of the soil.