Phase-sensitive measurements on a composite ring made of a superconductor of interest connected by a known singlet s-wave superconductor can unambiguously determine its pairing symmetry. In composite rings with epitaxial β-Bi_{2}Pd and s-wave Nb, we have observed half-integer-quantum flux when Nb is connected to the opposite crystalline ends of β-Bi_{2}Pd and integer-quantum flux when Nb is connected to the same crystalline ends of β-Bi_{2}Pd. With ascending temperature, the half-integer-flux quantization transits to integer-flux quantization, before the eventual loss of phase coherence. These findings point to odd-parity pairing symmetry in superconducting β-Bi_{2}Pd.