Generalization is the core objective when training optimizers from data. However, limited training instances often constrain the generalization capability of the trained optimizers. Co-evolutionary approaches address this challenge by simultaneously evolving a parallel algorithm portfolio (PAP) and an instance population to eventually obtain PAPs with good generalization. Yet, when applied to a specific problem class, these approaches have a major limitation. They require practitioners to provide instance generators specially tailored to the problem class, which is often non-trivial to design. This work proposes a general-purpose, off-the-shelf PAP construction approach, named domain-agnostic co-evolution of parameterized search (DACE), for binary optimization problems where decision variables take values of 0 or 1. The key novelty of DACE lies in its neural network-based domain-agnostic instance representation and generation mechanism that eliminates the need for domain-specific instance generators. The strong generality of DACE is validated across three real-world binary optimization problems: the complementary influence maximization problem (CIMP), the compiler arguments optimization problem (CAOP), and the contamination control problem (CCP). Given only a small set of training instances from these problem classes, DACE, without requiring domain knowledge, constructs PAPs with even better generalization performance than existing approaches on all three classes, despite their use of domain-specific instance generators.