As the low-density parity-check (LDPC) code has a powerful error-correcting performance and can achieve high throughput, it is being used in many application areas and recently adopted as a channel coding method in the 5G New-Radio communication standard. Unlike other LDPC codes, the 5G LDPC code has various irregular lifting sizes to support diverse message lengths. To meet the demanding requirements of the 5G standard, many solutions have been presented, but all of them are either impractical or fail to satisfy all the requirements. This paper, for the first time, proposes an area-efficient QC-LDPC decoder that satisfies the peak throughput requirements of the 5G standard and supports all the lifting sizes specified in the 5G standard. Instead of relying on full parallelism like in the previous works, this work tries partial parallelism to mitigate the hardware complexity, which leads to high efficiency in hardware complexity. In addition, a novel memory access scheduling method is proposed to solve the data access and alignment problems caused by the partially parallel structure, which is effective in supporting all the lifting sizes. A LDPC decoder realized in 65-nm CMOS technology demonstrates that its decoding throughput is greater than 20Gbps and its area is smaller than the existing decoders.