Recent years have seen increasing efforts to improve the input/output performance of multilevel flash memory. In this regard, we propose a coding scheme for two-page unrestricted-rate parallel random input-output (P-RIO) code, which enables different code rates to be used for each page of multilevel memory. On the second page, the set of cell-state vectors for each message consists of two complementary vectors with length n. There are a total of 2^n-1 sets that are disjoint to guarantee that they are uniquely decodable for 2^n-1 messages. On the first page, the set of cell-state vectors for each message consists of all weight-u vectors with their non-zero elements restricted to the same (2u-1) positions, where the non-negative integer u is less than or equal to half of the code length. Finding cell-state vector sets such that they are disjoint on the first page is equivalent to the construction of constant-weight codes, and the number of disjoint sets is the best-known number of code words in the constant-weight codes. Our coding scheme is constructive, and the code length is arbitrary. The sum rates of our proposed codes are higher than those of previous work.