Low-density parity check (LDPC) error correction decoders have become popular in diverse communications systems, owing to their strong error correction performance and their suitability to parallel hardware implementation. A great deal of research effort has been invested into the implementation of LDPC decoder designs on field-programmable gate array (FPGA) devices, in order to exploit their high processing speed, parallelism, and re-programmability. Meanwhile, a variety of application-specific integrated circuit implementations of multi-mode LDPC decoders exhibiting both inter-standard and intra-standard reconfiguration flexibility are available in the open literature. However, the high complexity of the adaptable routing and processing elements that are required by a flexible LDPC decoder has resulted in a lack of viable FPGA-based implementations. Hence in this paper, we propose a parameterisable FPGA-based LDPC decoder architecture, which supports run-time flexibility over any set of one or more quasi-cyclic LDPC codes. Additionally, we propose an off-line design flow, which may be used to automatically generate an optimized HDL description of our decoder, having support for a chosen selection of codes. Our implementation results show that the proposed architecture achieves a high level of design-time and run-time flexibility, whilst maintaining a reasonable processing throughput, hardware resource requirement, and error correction performance.