Conditional Optimal Transport (COT) problem aims to find a transport map between conditional source and target distributions while minimizing the transport cost. Recently, these transport maps have been utilized in conditional generative modeling tasks to establish efficient mappings between the distributions. However, classical COT inherits a fundamental limitation of optimal transport, i.e., sensitivity to outliers, which arises from the hard distribution matching constraints. This limitation becomes more pronounced in a conditional setting, where each conditional distribution is estimated from a limited subset of data. To address this, we introduce the Conditional Unbalanced Optimal Transport (CUOT) framework, which relaxes conditional distribution-matching constraints through Csiszár divergence penalties while strictly preserving the conditioning marginals. We establish a rigorous formulation of the CUOT problem and derive its dual and semi-dual formulations. Based on the semi-dual form, we propose Conditional Unbalanced Optimal Transport Maps (CUOTM), an outlier-robust conditional generative model built upon a triangular $c$-transform parameterization. We theoretically justify the validity of this parameterization by proving that the optimal triangular map satisfies the $c$-transform relationships. Our experiments on 2D synthetic and image-scale datasets demonstrate that CUOTM achieves superior outlier robustness and competitive distribution-matching performance compared to existing COT-based baselines, while maintaining high sampling efficiency.