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Neural architecture search (NAS) proves to be among the best approaches for many tasks by generating an application-adaptive neural architecture, which is still challenged by high computational cost and memory consumption. At the same time, 1-bit convolutional neural networks (CNNs) with binarized weights and activations show their potential for resource-limited embedded devices. One natural approach is to use 1-bit CNNs to reduce the computation and memory cost of NAS by taking advantage of the strengths of each in a unified framework. To this end, a Child-Parent (CP) model is introduced to a differentiable NAS to search the binarized architecture (Child) under the supervision of a full-precision model (Parent). In the search stage, the Child-Parent model uses an indicator generated by the child and parent model accuracy to evaluate the performance and abandon operations with less potential. In the training stage, a kernel-level CP loss is introduced to optimize the binarized network. Extensive experiments demonstrate that the proposed CP-NAS achieves a comparable accuracy with traditional NAS on both the CIFAR and ImageNet databases. It achieves the accuracy of $95.27\%$ on CIFAR-10, $64.3\%$ on ImageNet with binarized weights and activations, and a $30\%$ faster search than prior arts.