Pipeline parallelism is an essential distributed parallelism method. Increasingly complex and diverse DNN models necessitate meticulously customized pipeline schedules for performance. However, existing practices typically rely on predefined schedules, each with strengths, but fail to adapt automatically to the emerging model architectures. Exploring novel high-efficiency schedules is daunting due to the enormous and varying schedule space. Besides, manually implementing schedules can be challenging due to the onerous coding burdens and constantly changing needs. Unfortunately, existing frameworks have limitations in automated schedule exploration and lack flexibility and controllability. This paper presents FlexPipe, a programmable pipeline parallelism framework with enhanced productivity, programmability, debuggability, and ease of tuning. FlexPipe has two main components: a succinct domain-specific language (DSL) and an automated scheduler. FlexPipe enables automated schedule exploration for various parallel scenarios within a broad spectrum of schedule types at a small search cost. Besides, users can swiftly develop and customize schedules using the FlexPipe DSL, which embodies flexible controllability in the pipeline order of micro-batch computations over stages. It also provides convenient mechanisms to include new operations in schedules to meet changing demands. Our evaluation results demonstrate that FlexPipe achieves up to 2.28X performance speedup compared to the popular large-scale parallel framework Megtron-LM, and gains up to 1.49X performance speedup compared to the state-of-the-art automated pipeline parallelism framework.