Partial differential equations (PDEs) underpin predictive modeling in computational science and engineering, yet numerical computation of solutions can be prohibitively expensive for large, multi-physics domains, inverse problems or real-time control. Machine learning surrogates have emerged as promising techniques, particularly based on operator-learning frameworks. However, these methods typically rely on black-box regression of the solution operator and therefore struggle to generalize across complex geometries, enforce physical conservation, and respect boundary conditions. With growing interest in \emph{physics foundation models}, there is a need for principled learning techniques which address these issues by embedding physical structure to enable massive generalization. In this work, we present a geometry-conditioned framework for learning parametric PDE models on general meshes built on the Conditional Neural Whitney Form framework. We leverage data-driven finite element exterior calculus (FEEC) to enable learning geometry-adaptive reduced finite element spaces and nonlinear constitutive flux operators, while the governing conservation law, boundary conditions, and solution procedure are retained in a Galerkin compatible form. This results in a real-time reduced FEM model which can be efficiently solved to generate a predicted solution. This framework enables flexible use of current machine learning methods such as transformers for learning the encoding. Our framework integrates non-invasively with standard finite element machinery. The proposed approach is demonstrated on examples across complex geometries and large-scale problems in heat transfer, solid mechanics, and fluid mechanics, showcasing strong generalization capability and efficient computation. We record state-of-the-art performance on several standard benchmarks, highlighting the potential of structure-preserving learning compared to black-box point-cloud-based methods. The solve remains stable and incurs small additional cost. We additionally show the ability of the proposed framework to generalize out-of-distribution over domain sizes, topological characteristics, and other features. This work extends data-driven FEEC models to the geometry-general regime, providing a principled pathway for mesh-based scientific machine learning that combines the expressive power of transformers with the reliability and interpretability of reduced finite element models.