A photonic microwave downconverter with improved conversion efficiency and spurious-free dynamic range (SFDR) is proposed and experimentally demonstrated based on a dual-parallel Mach-Zehnder modulator (DPMZM) and a digital signal post-processing algorithm. The radio frequency (RF) signal and the local oscillator (LO) signal are fed to the two sub-MZMs of the DPMZM, respectively, leading to an infinite isolation between the RF and LO ports. By biasing the two sub-MZMs and the parent MZM of the DPMZM at the minimum transmission point, the optical carrier can be greatly suppressed. As a result, the conversion efficiency is improved for the same power impinged on the photodetector. A preliminary experiment shows that a conversion efficiency of -12.7 dB can be achieved. On the other hand, without emulating the inverse link transfer function, where exact parameters of the photonic link should be known, a simple post-processing algorithm which just needs the modulation index of the LO signal is employed to suppress the intermodulation distortion products. The SFDR of the downconverter is improved from 101.5 dB · Hz ^2/3 to 114.5 dB · Hz ^4/5 by using the proposed digital linearization algorithm.