We present a high-precision atom-interferometric test of the parity-odd spin- and velocity-dependent (SVD) interaction between the spin-polarized proton and unpolarized nucleons. The test utilizes the Bragg atom interferometer loaded with ^{87}Rb atoms, of which the single unpaired proton within the nuclei plays the role of the test spin. The differential measurement of the acceleration of the atom in two well-chosen inner states is designed to eliminate the influence from the polarized electron in the ^{87}Rb atom. Moreover, the atom interferometer is particularly placed in the cave laboratory within the Yujia Mountain in the campus, and the mountain source of unpolarized nucleons allows to improve test of the SVD on the length scale around 5-100 m. Our experiment improves the test precision of the universality of free fall to 9.2×10^{-9}, which is about 10 times better than the previous experiment with polarized atoms. More importantly, it provides a new constraint on the coupling of the SVD interaction exerting to the spin-polarized proton |g_{A}^{p}g_{V}^{N}|≤6.5×10^{-32} at λ=10 m, resulting in a substantial sensitivity improvement over the previous limit. Our work extends the scope of atom interference measurements and shines a new light on the testing of new physics with polarized-atom interferometers.