We report the first mass and distance measurements of a caustic-crossing binary system OGLE-2014-BLG-1050 L using the space-based microlens parallax method. Spitzer captured the second caustic crossing of the event, which occurred ~10 days before that seen from Earth. Due to the coincidence that the source-lens relative motion was almost parallel to the direction of the binary-lens axis, the fourfold degeneracy, which was known before only to occur in single-lens events, persists in this case, leading to either a lower-mass (0.2 and 0.07 ${{M}_{\odot }}$) binary at ~1.1 kpc or a higher-mass (0.9 and 0.35 ${{M}_{\odot }}$) binary at ~3.5 kpc. However, the latter solution is strongly preferred for reasons including blending and lensing probability. OGLE-2014-BLG-1050 L demonstrates the power of microlens parallax in probing stellar and substellar binaries.