In this paper we present an all-fiber interferometric sensor for the simultaneous measurement of strain and temperature. It is composed of a specially fabricated twin-core fiber spliced between two pieces of a single-mode fiber. Due to the refractive index difference between the two cores in a twin-core fiber, a differential interference pattern is produced at the sensor output. The phase response of the interferometer to strain and temperature is measured in the 850-1250 nm spectral range, showing zero sensitivity to strain at 1000 nm. Due to the significant difference in sensitivities to both parameters, our interferometer is suitable for two-parameter sensing. The simultaneous response of the interferometer to strain and temperature was studied using the two-wavelength interrogation method and a novel approach based on the spectral fitting of the differential phase response. As the latter technique uses all the gathered spectral information, it is more reliable and yields the results with better accuracy.