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In this paper, we present a new class of t-erasure horizontal codes with independent parity symbols based on Column-Hamiltonian Latin squares (CHLS). We call the codes PIHLatin (parity independent horizontal Latin) codes. We prove the necessary and sufficient condition of the existence of PIHLatin codes for t=2. For tges3, we prove some necessary conditions of the existence of PIHLatin codes. We also prove the bijection between 2-erasure PIHLatin-like codes and CHLSs and prove the mapping from t-erasure PIHLatin-like codes to t-1 mutually orthogonal CHLSs for t>2. The performance analysis shows that PIHLatin codes are superior to other multi-erasure array codes in flexibility and variety. Moreover, PIHLatin codes are suitable for both traditional disk arrays and distributed storage systems.