We report on efficient inscription of fiber Bragg gratings (FBGs) in a new type of single mode fiber with nanostructured core and with an effective parabolic graded index profile, using the standard phase mask method and a 248 nm pulsed laser. A nanostructured core allows to obtain high concentration of GeO2 in subwavelength glass rods and simultaneously to maintain low average germanium dopant level of silica similarly to standard single mode fibers. We showed that in a nanostructured core fiber, a factor of 3 better efficiency in gratings inscription was achieved, although the fiber has 20% lower average concentration of GeO2 with respect to SMF-28. In the nanostructured fiber we obtained a significant improvement in temperature sensitivity, while the strain sensitivity of FBG in nGRIN optical fiber is the same as in case of standard single-mode fiber (SMF-28). We have measured the strain sensitivity of 0.72 × 10−6 1/με (1.11 pm/με@1.53μm), and the temperature sensitivity is about 30% higher than for FBG in SMF-28 and equals to 10.2 × 10−6 1/K (15.6 pm/K@1.53μm).