A parabolic-cylinder-like long-period fiber grating (PCL-LPFG) is designed and experimentally fabricated successfully by a CO2 laser for the first time, to the best of our knowledge. The PCL-LPFG consists of an array of nonplanar PCL-shaped grids. The theoretical model of the PCL-LPFG shows that the PCL grid enables enhancement of the refractive index (RI) modulation efficiency and changes the mode coupling characteristics. The experimental results show that the strain sensitivity is 1.6 pm/με in the range of 0-3010 με, which is fourfold compared to the conventional LPFG, and its RI sensitivity is as high as 1422.69 nm/RIU and 456.44 dB/RIU, respectively. The sensitivities of the PCL-LPFG can be further improved by optimizing the characteristic parameters of the grating. Furthermore, the resonant wavelength of the PCL-LPFG can be continuously tuned by temperature.