Slender structures play a pivotal role in various engineering applications, exhibiting pronounced geometric nonlinearity. High-fidelity models based on geometrically exact beam theory are computationally expensive for multi-query tasks. Projection-based reduced-order models (ROMs) reduce cost but rely on modes extracted under fixed boundary conditions, making them incompatible with boundary variations. Existing methods (penalty functions, Lagrange multipliers) often suffer from ill-conditioning and require empirical tuning. This paper proposes the DEGP (Dirichlet-Embedded Geometric Parameterization) framework for geometrically nonlinear analysis of slender structures. The framework eliminates boundary constraint incompatibility by constructing interpolation functions for generalized coordinates, with Dirichlet boundary conditions embedded as interpolation data. It comprises four steps: (1) Parametric domain construction: A suitable parametric coordinate is selected and the structure is sparsely discretized into control points (including internal control points and boundary points). (2) Generalized coordinate selection: Generalized coordinates characterizing the dominant deformation are selected at control points. (3) Configuration function generation: Boundary conditions are translated into generalized coordinates at boundary points, and smooth interpolation (e.g., splines) connects boundary points with internal control points, generating a configuration function that a priori satisfies all prescribed Dirichlet boundary conditions. (4) Variational solution: Geometric coordinates of the current structure are obtained by densely sampling the configuration function, from which generalized strains and the total potential energy are computed; equilibrium reduces to unconstrained minimization of the potential energy. The method is validated on two-dimensional simply supported beams and three-dimensional helical structures under large deformations, with high-fidelity finite element solutions as benchmarks. By eliminating penalty terms, DEGP yields a better-conditioned problem with faster convergence, achieving less than 2% relative error with only 5–8 generalized coordinates, providing a promising tool for rapid prediction and inverse design of slender structures.